Finally something I can answer and provide a source for, although I'm not a historian, just an avid reader, sorry if I don't follow some of the guidelines(if I do, allow me to fix it). My main sources are The executioner in late medieval French culture and Discipline and Punish, there are a few facts that may have come from other books I'm not remembering the names of, and later I'll try to check back and fill their names if I'm able to find.

So, to the answer. There were many issues with death sentences before the guillotine, the main ones:

• Cruelty of the sentence;

• Complexity of executing the sentence;

• Inefficiency of some execution methods;

• Inequality of the sentences;

• Religious background of sentences;

• Expenses and cleaning.

Now the explanation. Before the guillotine, most death sentences in the West were given by royalty or the Church and they could vary wildly depending on who was guilty and what for. Sorcery could be punished by fire, boiling and a series of other cruel painful deaths, while nobles would mostly die by being decapitated by a sword, sometimes having some other member severed in resonance with their crime(an arm or even intimate parts in case of thievery or fathering a child with the wife of someone with great influence). The seemingly arbritary methods used gave a sense of Inequality and many times of unfairness to the punishments. Watching someone screaming and struggling in absolute pain for several minutes, watching blood boiling, even if it is a Witch burning, causes people to have doubt on the fairness of the trial(even more when they feel they could somehow end up there by mistake).

Also, when it came to efficiency, even if some of the executions were "simple", it is not as simple as Hollywood likes to make it seem. Swords could fail to complete the cut, damaging the target's neck without giving him/her a quick death, making them suffer. The most famous sentence, hanging, was slow and, if the neck didn't break and instead the victim died by asphixiation, the punished's blood would go over his throat and they would struggle like a fish out of water until finally dying drowned in their own blood. There was some even more complicated stuff, if you read the very first page of Discipline and Punish, you will find a description of a dismembering by horses, where there were necessary more horses than first thought (4 instead of 6, if I'm not mistaken), and still the process took many more hours than expected and had to be aided by axes and knives. Spilled blood, boiled corpses, bones, organs, nerves, there was a lot to be cleaned, that took time and was expensive. Know what also was expensive? Tools, executioners, scriveners, horses, safety measures and so many other things and people needed for executions to work, and all that had to be there through almost a whole day(if not more, in rare cases).

The guillotine was seem as much fairer, much more humane, an equal death to everyone, no matter the crime, no matter the origin. It was also much less messy, incredibly quicker, almost inexpensive and much more reliable. It solved most of the issues previously found on the other sentences, it was a wonderful advancement.

Edit: Thanks to everyone that appreciates my answer. Still, I think it is too crude, so it would be great if anyone could provide more details, answer the subsequent questions and provide more sources.