As other answers already say, hanging out sounds less childish than play.

There's a slight difference in meaning, though, too.

The word play often connotes playing around with some toy or game: playing marbles, playing with dolls, playing with Legos, etc.

The term hang out refers to spending time together, but not necessarily doing anything in particular: hanging out at the mall, hanging out after school.

When kids are hanging out, they are probably just standing and talking – usually at some location away from home. If they are playing, then that gets associated with playing some game or sport.

Another key is that play sounds childish by itself. Much like in the movie, a teenager is susceptible to being mocked if he says, "Anybody wanna play?" However, that same teen could probably get away with something like, "Anybody wanna play basketball?"

I haven't seen the film (or read the books), but based on your description of the scene, I would say that the other kids mock the "wide boy" for using play because it has more of a childish connotation than hang out. Both terms have the same general meaning of people spending time together while engaged in some form of social activity, but hang out is the "cooler" of the two and would be more appealing to kids once they've reached an age where they start wanting to feel more mature (i.e. "grown up").

By definition, there isn't much difference. Hang out is a term used by teens, play is a term used by kids. Hang out doesn't involve that much movement, maybe like chatting. Playing is playing a game or something.