What does the player role 'Captain' in FIFA 14 do?

No, it won't change gameplay in any way. Basically, Captains in FIFA only exist so that it's close to real football.

And while in real football, the captain is usually a player with a good tactical awareness, and someone who can lead the team, to whom everybody sort of listens, in FIFA it only comes down to preference, nothing more.

I normally choose a player who seems most like a captain to me. In Career Mode (I manage Atlético Madrid) my captain is David Luiz, which is simply because I noticed that in games he is often a driving force, providing assists to strikers and running into free space. Lately though, I have been experimenting with giving the armband to other players, like Javi Martínez in Midfield, or Hulk as a Striker. And it didn't make one tad bit of difference! So as I said before, you can pick whoever player you think looks best shaking hands with the opposite captain or whatever :) It really doesn't influence gameplay at all.