From what I understand,
試合 appears to apply mostly (only?) to sports, but is also acceptable and considered correct for board games (chess, 将棋【しょうぎ】, 碁【ご】, etc. - see comments below).
勝負 applies to these situations as well, but also to "fights"/"fighting" (actual, struggle for power, etc.), and to "the game" as in winning the game (the victory). The characters for
勝負 literally mean "win lose", so it can also have that definition.
- 真剣勝負をする → Play/Fight for real
- 勝負の世界 → The world where power rules
- 勝負に出る → Make a decisive move
- 勝負はこっちのものだ → The game is mine! (as in "I'm going to win!" or "The victory is mine!")
- 勝負は時の運だ → Victory or defeat [Whether you win or not] is just luck.
Here is an image from the game Samurai Shodown (サムライ・スピリッツ) showing the pre-fight announcement ("Fight!").
And another picture from the same game when you win the round (
勝負あり - "You win")
They also differ when used as
試合をする simply means to "have/play/hold a game/match". But
勝負をする can take that meaning as well as "to fight/challenge".
- 正々堂々勝負をしよう → Let's play fair [a fair game].
- 実力で勝負する → Dictionary says "match one's skill (against a person)"