It mostly comes down to 3 reasons:

1. Camels are efficient pack animals.

2. Roads are expensive.

3. Wheeled vehicles don't always interact well with narrow and crowded urban streets.

Basically, you can make cross-country wheeled vehicles, but these need large wheels for off-road use. For efficient transport of goods, off-road wheeled vehicles tend to be large, and before motorisation, typically used large teams of horses or oxen. They are less efficient off-road than on good roads (needing more traction force, and thus a larger draft team). Soft ground and steep terrain can defeat them. They are often unusable in city streets.

Given not-too-rough and not-too-steep hard ground, wheeled vehicles can be more efficient than pack animals. With a good road network, they are often the best choice, economically, taking into account the cost of the vehicle, the draft team, and the humans. However, roads need to be built and maintained. Including this cost, the wheeled vehicle might not be more economically efficient.

Whether wheeled vehicles or pack animals are the best choice depends on multiple factors: population density (how many people are available for road maintenance and to pay taxes for road maintenance), traffic density (good moved for a given expense on road maintenance, and again, taxes if the traffic is taxed), the pack animals available, and the terrain. An added consideration is whether the whole length of the road is under the control of a single state - if multiple states control the road, and some don't maintain the road, part of the road can be unusable, making pack animals necessary for part of the journey. There is a further maintenance-related factor: wheeled vehicles usually cause more wear-and-tear on roads than pack animals or porters (which is why some road networks banned wheeled vehicles).

For urban use, wheeled vehicles need space (including turning space, which can be a problem for 4-wheeled vehicles). Stairs present significant difficulty (consider the pre-1988 Dalek-stair problem). Pack animals and porters can work better for transporting goods in urban environments than wheeled vehicles (but some wheeled vehicles manage better than others, like traditional Chinese "wheelbarrows"). Palanquins and riding horses, and simply walking, can be better solutions for moving people.

The reduction in the use of wheels across much of Western and Central Asia came in two steps: first, when horse riding became common, the chariot declined. Second, when the camel came into widespread use as a pack animal, it was often the best economic choice for cross-country transport.

Japan also saw little use of wheeled vehicles, until the rickshaw and, shortly afterwards, motorisation. The Tokugawa government banned wheeled vehicles on major roads. This reduced road maintenance costs, and was perhaps also intended to protect the jobs of porters. It has been argued that the ban was to enhance government control - e.g., by u/AsiaExpert in https://www.reddit.com/r/AskHistorians/comments/17blcv/ive_heard_in_medieval_japan_wheeled_transport_was/ - but moving transport from foot and horse to wheels restricts it to roads, makes it harder to cross rivers without bridges, etc. and would appear to make travel easier to monitor. Wheeled vehicles remained in local use, off major roads, and were usually human-powered: wheeled shrines for religious processions, and night soil carts.

Africa can be divided into multiple cases. North Africa connected into the Western Asian world, and similar factors were at play. The Mediterranean also offered efficient bulk transport. This is also true of the Nile, and might explain the very slow adoption of wheeled vehicles by Egypt, which largely ignored the use of the wheel in nearby Mesopotamia for well over a thousand years, until conquest by chariot-using armies. The Sahel, while less connected to Asia, appears to have offered similar economic choices, and ancient use of wheeled vehicles disappeared in favour of pack animals and riding horses. Parts of sub-Saharan Africa have difficult terrain and tsetse fly can made draft animals infeasible (and also pack animals, which resulted in human porters being the best transport choice when away from rivers).

The Americas don't appear to have adopted the wheeled vehicle at all, despite having wheels (e.g., wheeled toys) and used water transport and porters.

Finally, the kinds of loads matter. If only light loads need to be moved, then they can be carried by porters or pack animals. Large heavy loads that can be broken down into light loads can also be carried by porters or pack animals. Large loads that can't be broken down in small loads suitable for pack animals (e.g., huge stone blocks) can need to be moved by heavy wagons, or boats, given rivers, lakes, or oceans (or, depending on the load and the distance, can be dragged or moved on sleds or rollers).

For more on the replacement of wheeled vehicles by camels etc., see

• Bulliet, Richard W., The Camel and the Wheel, Harvard University Press, 1975.

• Bulliet, Richard, The Wheel: Inventions and Reinventions, Columbia University Press, 2016.

It was Richard Bulliet in his 1975 The Camel and The Wheel who first called attention to and attempted to explain the fact that wheeled vehicles were almost entirely absent from the Middle East and North Africa in the Medieval and Early Modern periods. His book is still taken as the touchstone for this topic so I'll be basing this answer on it. Bulliet's argument is fundamentally economical: in the early centuries of the first millennium, technological changes in the nature of camel harnessing made it much cheaper to utilize them for transportation in comparison with wheeled vehicles. As camel transportation became increasingly prevalent there was a corresponding decrease in the need to maintain road networks suitable for wheeled vehicles, further incentivizing animal power. The result was that the art of wheel- and wagon-crafting disappeared from the Middle East and North Africa, regions where the camel came to predominate. As for why camels (once properly harnessed) were superior for the job, I'll reproduce here his list taken from a report on the superiority of camels over ox transport for military purposes by the 19th-century British officer Major Arthur Glyn Leonard:

1. [Camels] can carry or draw twice as much.

2. Faster, and able to cover more ground daily.
3. Can do from 20 to 25 miles in one stretch.
4. Will make many more journeys in a year and in their respective lifetimes.
5. Able to traverse ground that a wagon will stick in.
6. No trouble fording rivers, where wagons would have to be unloaded.
7. [not germane]
8. Live and work four times as long.
9. Greater powers of abstinence from food and water.
10. Greater tenacity and endurance.
11. Wagon liable to break down, upset, or stick. Consequent loss of time and additional expense in former case, besides inability or want of means to repair.
12. Lastly, the additional dead weight of the wagon, which is considerable - at least a ton, I should say.

Camels were hardy, versatile, and cheap - once they could be effectively harnessed for transportation purposes, and once they became widely available. Bulliet links this to the invention some time between 500 BCE and 100 CE of what he calls "the North Arabian Saddle," a contraption that allowed rider to perch on top of a camel's hump in a way that distributed his/her weight around the hump and to the camel's ribcage, or in the case of loads for transportation, that distributed weight evenly onto both sides of the saddle. This allowed both for more economical use of the camel as a pack animal as well as its effective use in warfare, and contributed to an increase in the military power of Arabian tribal groups - and consequently, to a growth in the availability of the camels that they raised to the settled societies of the Fertile Crescent. The process by which camel transport displaced wheeled vehicles thus began prior to the Islamic Conquests of the 7th century, but the latter probably helped push the process along.

In this context, the boundaries of this "wheelless zone" extended to wherever large-scale camel raising and trading was made viable by climatic, economic, and political conditions. Arabian camels couldn't thrive to the same degree in Anatolia's generally colder climate, and even after the Turkish conquest Anatolia never developed the critical mass of camel-rearing pastoral nomads needed to drive the price of the animal down to a level that could devastate the market for wagons (sheep and horses were far more popular). The camel's ability to beat the wheel lay not just in that animal's ability to out-compete wagons in an absolute sense, but in the wider climatic and economic conditions in the Middle East that made it cheaper there than elsewhere.

So in general, the received wisdom is that camels largely supplanted wheeled carts and wagons in the Middle East in late antiquity due to political/social (the increased prominence of desert nomads) and technological (new and superior saddles) reasons and perhaps also the introduction of sturdier camel breeds. The argument comes mainly from Bulliet’s 1975 The Camel and the Wheel and has been fairly widely heeded; camels are capable pack animals, which can carry considerably heavier loads than most other pack animals and offer other environmental advantages in the Middle East, so it’s not an unattractive argument. You’ll find Bulliet’s general thesis presented in summary form in generalized reference works for late antiquity, such as the “camel” entry in the encyclopedic Late Antiquity: A Guide to the Postclassical World (1999), which states in part the following:

"Once Arab prosperity and enterprise had brought the desert nomads into closer relations with the settled peoples of the surrounding lands, opportunities arose to capitalize on the camel’s strength and cheapness. Diocletian’s edict on prices of 301 C.E. indicates a 20 percent savings for hauling a load on a camel instead of an oxcart, and there are other indications that camel breeders took over much of the transportation economy. As a consequence of this, and perhaps other factors, wheeled transportation virtually disappeared from the Middle East by the time of the Islamic conquests, except in Anatolia, which was probably too cold for the Arabs’ camels. Carts proved slightly more enduring in Egypt and North Africa, but they mostly succumbed to the camel competition there too."

I can add that the technology of wheeled transportation has other social costs besides those directly reflected in the immediate cost of paying someone to move your stuff: roads, for example, were expensive both to build and to maintain in good repair, and pack animals can do without them more easily than wheeled vehicles.

Lastly, wagons may be able to carry more, but they were a pretty big capital investment compared to pack animals and potentially not worth the considerably larger cost. As Bagnall pointed out in The Camel, the Wagon, and the Donkey in Later Roman Egypt (1985),

"First, the Egyptian countryside was extremely cut up with canals, drainage ditches, dikes, and other irrigation works. Animals are easier to get around with than wagons in such an environment. Secondly, wagons are expensive, camels cheaper, donkeys cheapest."

Bagnall doesn’t actually get to endorsing Bulliet’s Middle-East-wide grand thesis for the specific (and in many respects special) case of Egypt – making the point that the camel had competed with the donkey and the wagon in Egypt for many centuries and “drove neither out of use,” though he also allows that “the wagon, never in very common use, became still less common.”

Overall, the camel-centric argument does seem to have a few thin points – in particular that it doesn’t really consider the continued, fairly prominent role of other pack animals, such as the aforementioned donkeys – and attempts have been made to supplement it by pointing out that, e.g., cultural factors as well as camel competition may have been at play (for example, that the continued use of wagons in Anatolia may have also had to do with a cultural history of wagon dwelling on the steppe before the Turks’ migration into the area). Similarly, one can propose that pack animals are more attractive in sparsely populated areas that may not have the means for roads (but then, oxcarts were once used in the Sahara, and largely stopped being used even in prosperous Syria). Overall, though, the basic argument is likely to continue to be that the advantages of wheeled transport just may not have been all that clear cut in comparison to the technology’s relative costs.

p.s., KOM, I know you from gaming! Ethiopia rules the waves!

p.p.s. edit: I see I am a little late to this party!