They're related concepts, but I'd say that they won by building a superior product for a niche.
I first got on FaceBook in the fall of 04, when they were still "elite colleges only", and first checked out MySpace around that time too (I did end up creating a MySpace profile, but abandoned it after like half a dozen visits). MySpace was better as a general social-networking site. It had all sorts of features that FaceBook didn't - you could put pictures and videos and music on your profile, it had messaging (FaceBook only had the Wall when it launched, which was everyone-to-everyone messaging and lost the thread of conversation), and it had all these features for bands and other local entertainment.
FaceBook was better at one, specific use case: you meet someone at a party, you want to remember their contact details so you can hang out with them again, you track them down on FaceBook. And it was a lot better than MySpace at this: your profiles were private to within the school so they wouldn't be exposed to the whole Internet, and you could easily search within schools, and you could find their contact info & interests at a glance without having to wade through all the crud people put on their profiles. And that one use case turned out to be very useful for very many college students.
That's an interesting perspective about the use case - I signed up at about the same time, never use the account and deleted it a few years later. I was more the steady girlfriend type than the party type in college, so perhaps that use case passed me by.