The Winklevoss subplot was to show Mark's motivators. The Winklevi were the thing he was trying to beat--clubs that wouldn't let him in. The very first scene shows how paranoid Mark is about his girlfriend liking Big Elite Rich Athlete Clubs. This pushes him to try to get both revenge on her and to win her back by showing he is better than those elite clubs.
There's several bits of the writing that show this. One is that, after he sees Erica at the bar and she says "Good luck with your video game". In the script, he has a hurt reaction shot, but the movie just makes him angrily storm off and tell Eduardo they need to expand--specifically to BU. Someone asks why BU, but he just moves onto a new topic...because he didn't want to say BU is where Erica goes.
Later on, when talking to Sean about Sean's old high school girlfriend, Mark asks if he ever thinks about the girl. Sean just looks incredulous and says no. But clearly Mark is...
All that leads up to the final shot of him trying to friend Erica, which plays into the movie's overarching theme of trying to win the girl back.