I will preface this by saying that I'm not a student of near eastern arms and armour, proper, and that I'm more a student of armour history than sword history, but I can speak to the metallurgy involved. Anyway, it's morning, I don't have to be into work until 11, an I put on Yes's Relayer, because if you're going to answer a question that contains the words 'Damascus Steel' you had better have an appropriate (self-aware) soundtrack. Let's do this thing.
First off, let's talk about steelmaking, because this is the difference between most pre-modern steel in Europe and 'Damascus' steel lies in how the steel itself was made. This in turn reflects the construction of the sword.
The first and simplest way to get iron out of ore (that can also make steel) is a bloomery. This is a simple furnace (generally rather tall) that combines iron ore and charcoal in layers. When the charcoal is lit, it burns and burns off the non-iron compounds from the ore (a lot of iron ores are mostly iron oxide, so in a furnace the oxygen becomes part of the combustion of the charcoal - to simplify a lot, you start with carbon and iron oxide and are left with iron and carbon dioxide). Other impurities hopefully melt out and pool at the bottom as a kind of glass, slag. The iron in a bloomery doesn't quite melt but it kind of globs together into a spongey mass at the bottom, a bloom. Because it hasn't melted and there are still impurities, and because even the combustion process introduces other impurities (like silcates from the clay of the furnace) there is generally still slag in the bloom. This is generally manually worked out by hammering the bloom or folding the resulting metal over.
Now, in the right conditions a bloomery can also make steel - if there is a little less oxygen, some of the carbon from the combustion will combine with the iron to form iron carbides that mix with the iron in a variety of crystalline structures. This is steel, of low or medium carbon content (under 1%).
If you build a larger furnace that runs at higher temperatures, you can melt the iron (this happens when it absorbs enough carbon to lower its melting temperature, as well as when it gets hot enough). This is a blast furnace, which was developed first in china and then used in Europe sometime around the high Middle Ages in various places (Wallonia, possibly Scandinavia, a few others). However, blast furnaces produce cast (or pig) iron, which is very high in carbon content and is consequently quite brittle. It can be annealed by reheating to be a bit less brittle, but it's not the ideal material for swords, and wasn't used for them in medieval Europe. Cast iron is important for damascus steel, though - as an ingredient.
So this covers the basics of iron smelting and the simple, one-step process of steelmaking used in medieval Europe. What about the process used for 'damascus' steel?
To talk about damascus steel, we need to talk about crucible steel. Crucible steel is not a direct product of iron-smelting, but a way of taking the products of it and turning them into a better steel. In surviving Arabic recipes for crucible steelmaking, bits of bloomery iron are combined with either a carbon source or bits of higher-carbon iron (like pig iron from blast furnace processes) and melted in a vessel, a crucible, to produce a higher-carbon steel. Because it is melted with a source of carbon (of some kind) it is higher carbon than the steel produced by a simple bloomery process - over 1% carbon. This is called a hypereutectoid steel Because it is melted and then cooled, the resulting cake is decently sized (maybe a pound, maybe several) and hopefully relatively homogenous. Historically this process was probably developed in India and spread to central asia, and traded with the Arabs of Damascus who turned it into swords and knives. Properly speaking, damascus steel is crucible steel that has a visible 'watered' pattern (kind of a wavy look) - this is how the term came to be used for pattern-welded steel as well, but there is other crucible steel that is pretty similar for most metallurgical purposes.
Now it's worth noting that neither medieval European nor central/south Asian steelmakers knew the chemistry of what they were doing. They thought of smelting as refining iron ore with heat,which is accurate enough, but they also generally thought of steel as being a purer, more 'refined' type of iron, not as an alloy of iron and iron carbide. That carbon source in crucible steelmaking? It was often medicinal herbs, or pomegranate seeds, or something else with alchemical properties that were desired. This was craft and alchemy for the people doing it, not chemistry.
So how did these effect swordmaking? For medieval Europeans, until the late middle ages using bloomery steel meant that they often made swords out of multiple pieces of iron and steel. One way of doing this is taking several bars of iron or steel and twisting/folding them together. Ideally the folding process and the subsequent hammering and rehammering helps homogenize a rather heterogenous collection of metals. The most famous example of this is the pattern welded blades of Scandinavia (and the swordsmithing tradition of Japan, which also uses bloomery steel from a slightly different type of bloomery). Ideally, the edges should be hard so that they can keep an edge, so often higher carbon steel is forge-welded onto the edge of the sword, or the sword edge is case-carburized - heated with a source of carbon to increase its carbon content. Importantly, in the 13th century most European-made swords would be made of multiple pieces of metal forged together. Sword smiths were pretty good at this, and it is only by microscopic studies (or looking for pattern welding, in the case of scandinavian blades) that we can often tell that these swords aren't homogenous, and they look similar despite being made through a variety of processes of combining different pieces of metal. Moreover, because bloomery steel doesn't start out particularly hard, it needs to be hardened by a variety of heat treatment processes involving heating the steel, cooling it more or less rapidly, and sometimes reheating it. This can double or triple the hardness of a medium-carbon steel, but does nothing for pure iron, so this can produce swords that have different hardnesses - ideally, hard edges, if nothing else!
Crucible steel, by contrast, can be made into a relatively homogenous sword. If the ingot from the crucible is smaller, you may need more than one, but a sword made of crucible steel will be fairly hard throughout. Moreover, because it is so high in carbon it is already quite hard even without heat treatment - it is not unusual for an un-heat-treated crucible steel sword to be about as hard as a heat treated sword of lower carbon content! This homogenity and hardness means that crucible steel blades are great at keeping an edge. However, the craftsmanship of good European blades can make them very tough as well, so there isn't some vast superiority. Wills show that Carolinian Europeans prized good swords 'of India' and accounts from Al Andalus show that Moors prized good northern European 'Frankish' blades.
As to a direct comparison of cost, I am not aware of one. There are possibly a group of viking-era swords that are made from imported crucible steel, or perhaps from indigenous European crucible steel (a process that was discovered and then lost or discarded, perhaps), the 'Uthbert' swords. u/alriclogar talks about these here. But I'm not aware of any references to damascus steel swords in 12th or 13th century Europe comparing them to European blades in terms of cost etc.
Sources:
The Sword and the Crucible, Alan Williams
Arms and Armour of the Medieval Knight, Edge and Paddock
"Damascus steel" is used with multiple meanings, which affects the "unique" part of your question. The 3 main meanings are:
"Watered steel", steel that displays a pattern similar to watered silk. This appears to be a common historical meaning, and is also a common modern meaning. This includes both patterned crucible steels and pattern-welded steels; the latter are sometimes called "mechanical damascus" to distinguish them from the former. With this meaning, the question of "unique" is rather watered-down, since swords made with patterned and non-patterned crucible steels can be very similar in their mechanical/physical properties, and pattern-welded steels with watered patterns can be very similar to other pattern-welded or laminated steels. So, while this is a common historical meaning, it isn't a very useful one when it comes to your question: therefore, note this meaning and move on.
Ultra-high carbon crucible steels, including Indian and Sri Lankan wootz and Central Asian pulad or bulat. This includes both patterned and non-patterned crucible steels. This would also include East Asian crucible steels (if they existed, and they appear to have, before being displaced by Chinese steel made by decarburisation of cast iron). This is the most useful meaning for your question.
Indian crucible steel, wootz. This isn't unique, since it's mechanically the same as Central Asian crucible steels. Not so useful for your question.
So, sticking with meaning 2, what is unique about damascus steel? The carbon content is very high - typically between 1% and 2%. How does it get this very high carbon content? Wrought iron and a source of carbon (either plant material, which will be converted to charcoal as it's heated, or seashells (calcium carbonate), or cast iron (which has 3-4% carbon)) are put together into a sealed crucible, and heated, and kept hot for long enough for the carbon to diffuse into the iron. For wrought iron + cast iron, the cast iron (which has a significantly lower melting point than wrought iron) will melt, and this will help the process. If you want the whole thing to melt (i.e., the wrought iron + cast iron mix), you want to have enough carbon in the final product to drop the melting point low enough. This is an important reasons for the carbon content of the final crucible steel to be so high.
Is this ultra-high carbon content unique? No. Bloomery steels can exceed 1% carbon (but much higher, and one risks having the steel melt in the smelter, and giving a puddle of cast iron as the product, so typically not over 1.5%), and the Chinese steel-making process via decarburising cast iron could be easily used to produce ultra-high carbon steel.
The very high carbon content changes the conditions under which it can be forged. If forged correctly, it will result in a steel with small spheroidal carbide grains. These carbide grains are very hard, and result in improved wear resistance of the steel. The very high carbon content can also make heat treatment of the steel difficult - if it isn't done correctly, a sword made with this steel can be brittle. One historical method that was used to avoid brittleness was to not harden the steel, to let a blade slowly air-cool rather than quenching it in water. This results in a blade that is soft, reasonably tough, and can still maintain a good edge due to the wear resistance resulting from the carbide grains. Still, this potential for brittleness can make crucible steel less desirable for many uses.
Apart from wear resistance due to carbide grains, crucible steels have another advantage over bloomery steels (i.e., the steels used in Europe): they have a lower slag content. This means that the steel doesn't need to be repeatedly folded to get rid of excess slag when it is worked. The slag content is lower because the wrought iron that is used to make the crucible steel has already been repeatedly folded to get rid of slag, and if the carbon source is cast iron, that has less slag to begin with since the slag is separated from it when it is molten (unlike bloomery steel/iron, which doesn't melt during smelting, and traps the slag inside it). While bloomery steel can be folded to get rid of slag, the carbon content typically decreases during the process, and the final carbon content can be lower than desired. Crucible steel can also be usefully folded: the potential brittleness can be mitigated by folding the steel, not to get rid of slag, but to get rid of excess carbon.
How well does crucible steel work for swords? Very well, if you do conservative heat treatment (like the traditional air-cooling rather than quenching). The final sword will hold a good sharp edge due to wear resistance from the carbides, and won't be too brittle, and will have reasonable good springiness (i.e., if bent, it will return to the original shape, rather than staying bent).
How does this compare to pattern-welded swords? The best pattern-welded swords appear to be better. They have harder edges, and are unlikely to break. Even if the hard edge cracks, the soft body will support it. A pattern-welded swords with an iron (rather than steel) body will be very tough, but if it is bent, it is more likely to stay bent than a crucible steel sword (but can be bent back into shape later). If the carbon content of the steel used for the edges isn't high enough (high enough would be about 0.6-0.8% carbon), the edges won't be much harder than the edges of an air-cooled crucible steel sword, and won't have as good wear resistance. Such a sword would still be perfectly functional, but not as good as a typical crucible steel sword or a better pattern-welded sword.
"Toledo steel" swords were typically laminated blades. An iron or low carbon steel body with a high carbon steel skin (like Japanese kobuse lamination often used on Japanese swords) was common. (This kind of lamination was common on Medieval and post-Medieval European swords, not just swords from Toledo.) Functionally, this will give a sword similar in performance to good pattern-welded swords, as long as the heat treatment (quenching to harden and tempering to reduce brittleness) is done well.
It's hard to compare crucible steel and other swords directly. I don't know of any 12th to 13th century comparisons. Earlier, we have al-Kindi's book on swords ( (original text, translation, and commentary are available in Robert G. Hoyland and Brian Gilmour, Medieval Islamic swords and swordmaking, Gibb Memorial Trust, 2006) discusses various types of swords, including pattern-welded blades and blades made from crucible steels - the pattern-welded blades are of similar quality as good crucible steel blades (and good Middle Eastern bloomery steel swords). Al-Kindi notes that the European pattern-welded swords are less brittle in cold weather.
In the 14th century, Ibn Hodeil, in Equipment for Knights and Insignia of the Brave, wrote: "The Arabs claim that the fabrication of (Frankish swords) is the work of Genies. They are more resistant to blows one gives with them even during cold weather, while the hindy (wootz) sabre often breaks when the weather is cold" (as quoted in https://www.tf.uni-kiel.de/matwis/amat/iss/kap_a/advanced/ta_4_2.html (bracketed comments in that source)). So, both sides of the 12th-13th century, we find comments that crucible steel swords can be brittle in cold weather.
We have later comparisons, from European colonial experience in India. This includes comments that swords made from Indian steel are inferior because they are more brittle. Note also that many sword blades were exported from Europe (typically made in Solingen as trade blades for export), and these blades were highly regarded in India. In the 19th century, British cavalry sabre blades were recycled as tulwar blades, and were considered excellent blades by their Indian users.
Another comparison is the various Ulfberht swords. The (probably) crucible steels ones are often high-quality, but some have poor heat treatment, and some have broken. The good pattern-welded ones are also of high-quality, and typically have harder edges than the crucible steel Ulfberhts. For discussion of these, see Alan Williams, The sword and the crucible, BRILL, 2012.
So, in summary, crucible steel swords can easily be better than low quality swords (pattern welded and laminated swords with low carbon steel, and iron swords), and good crucible steel swords will be of similar quality to good laminated and pattern welded swords. However, crucible steel swords can be brittle.
As for prices, al-Kindi comments on prices of different types of crucible steel swords, but doesn't say anything about non-crucible-steel swords. The prices he does give vary from under 1 dinar for inferior swords to 50-100 dinars for high quality swords. Indian crucible steel, in India, was relatively cheap steel, and was widely exported. One reason why it was cheap was the ready availability of wood (for charcoal, as fuel and also as a carbon source for smelting). As it was travelled further after export, the price increased. Where it remained cheaper than locally-made steel, it could be a popular import item. In at least some places, it doesn't appear to have been considered significantly better than local steels - price determined its desirability. It certainly isn't "magic steel", better than any contemporary alternatives.
For lots of info, both technical and historical, on steel and swordmaking, perhaps the best online resource is Helmut Föll's Iron, Steel and Swords:
https://www.tf.uni-kiel.de/matwis/amat/iss/index.html
For crucible steel, see 10.4. For crucible steel swords, see 11.5. For a discussion of primary sources on crucible steel, see https://www.tf.uni-kiel.de/matwis/amat/iss/kap_a/advanced/ta_4_2.html (as cited earlier). For Ulfberht swords, see 11.4.3.
While you are waiting for new answers, you may be interested in this previous answer by u/wotan_weevil