Actually the reason is purely psychological. When you put the limit on the length of the video you make it really easy to make bad quality videos. Of course, everyone can make a bad quality video on YouTube, but then it would compete against many good quality ones, and the user would feel pressure against uploading it. If all videos are short and simple, it's much easier to create and share one.
Thus, Vine vs YouTube is like Twitter vs LiveJournal. On LiveJournal there are thousands of great authors who write literate and beautiful and thoughtful posts. But this sets a very high barrier for new participants. After all, not everyone's writings are inspiring and thoughtful. Twitter came to the rescue: nobody can be bad on Twitter because nobody can be good either (ok, there are exceptions of course, but wait until MiniTwitter comes in with only 12 letters per message :). With Vine it's the same: shoot a short clip, then upload. No need to worry about post-processing, sound tracks, special effects, embedded cross-links, and such.
That was the production side. But there is also the consumption side (mentioned in other posts and comments).
Short videos are easier for your friends to watch, and thus can be done quicker -- for example the moment you share it, -- not some time later in the evening or maybe never, as would be with YouTube videos. The file size argument is actually much less important -- today's connection speeds allow me to upload a 10-min clip in just a couple of minutes, and even do that in background if I want to.
TL;DR: Vine is like video-Twitter.