Maybe it's a regional thing (I'm from the East Midlands), but playing with the team has a different connotation from playing for the team to me - although both are valid sentences.

Playing for the team suggests that I'm part of that team. Playing with the team would say to me that I'm not part of them, but I've been co-opted for a game or something like that.

Oddly, I would say "playing with the orchestra" rather than "playing for ...".