Difference between 'play you' and 'play with you'
Solution 1:
I feel that "I'll play you" is competitive, where as "I'll play with you" could be collaborative. So you could play someone at tennis, but you could play with someone in a doubles game.
Solution 2:
I'll play you next means that you will be my next opponent. I'll play with you next might also mean that, but generally it would suggest some less serious kind of ludic activity.