I've put together 4 computers from components.
Some unmentioned advantages:
Cosmetics are teased apart from performance.
The main thing you'll be "looking at" will be the case. The choices for cases are innumerable and you can put whatever you like inside them (as long as it's the right form factor, of course). If you like flashy LED stuff and see-through sides you can get a case with a window and LED lighting and other weird stuff. If you (like me) prefer something more simple, you can find many more standard and straightforward cases that don't try to be anything special but have excellent build quality. My point is, when building you don't have to consider appearance at the same time you consider performance- eg there is no "I like how this one looks, but this one is faster" consideration. The two are separate and you can choose how to balance them as part of your budget. You do not get that choice with a pre-built and honestly a lot of pre-built manufacturers use kind of ugly designs (which isn't to suggest that all bare cases are beautiful, either)
Another advantage to building it oneself is that you will be familiar with the insides. Upgrades tend to be easier in general, and most motherboards you can buy bare are more feature complete than prebuilt systems. Often with pre-built motherboards entire features are removed, stubbed out, or otherwise unavailable, and sometimes they will use non-standard connections, and this can leak out to add-on cards that otherwise come with the system.