A play is a piece of literature written for the theatre or television.

• Julius Caesar is one of Shakespeare's early plays.
  

A game is, for example, chess, football, or bridge.

• Chess is a very slow game. (NOT . . . a very slow play.)

Verbs: people act in plays or films, and play games.

• My daughter is acting in her school play. 
• Have you ever played rugby football?