So, you're at an ultimate frisbee game. Everyone introduces themselves. You can then go home and find them on Facebook based on their name and mutual friends (since you probably know some other people on the team). No asking for contact info and all the associated social baggage. There's one benefit.

Now, once you're friends with them, Facebook passively (or actively, with minor communication such as comments) maintains familiarity. Basically what this means is if you happen to run into them six months later, it isn't like meeting them for the first time. Facebook can by no means fully maintain this sort of low-level relationship indefinitely. However, I've found it's very effective at increasing the half-life of such things.

Facebook automates the maintenance of my social network. That is valuable.