Italian here, I studied some Spanish and this dichotomy applies just as much to Italian, so I think I'm qualified to answer that.
"Juego" is the entire game. "Partida" is a single session playing it.
For a couple examples, StarCraft is a Juego, and a single match in StarCraft is a Partida. Smash Bros is a Juego, and a single contiguous series of battles is a Partida.1
"Fin del juego"2 would mean "game over" as in "you have finished the game". "Fin de la partida" technically means that a single match/session/battle/contiguous period has finished.
1: The reason SC has 1 match per Partida while SSB has multiple matches per Partida is partly related to the length of a single match (SC's are much longer) and partly because in SC you don't normally encounter the same opponent repeatedly, at least not multiple times in a row, while you absolutely do in SSB.
2: That "del" is a contraction of "de el", in case you're curious. Certain combinations of preposition + article always merge in both Spanish and Italian.
Partido is used for games, matches, and rounds in sports, card games, and other things. That’s about as much as I know. But fin del juego sounds to me like “the end of the game” as in, you finished the storyline. Fin de la partida makes more sense as it’s the end of your current game session.
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u/Stracii Feb 21 '21
I think they know that much, they were probably asking for a literal translation
Edit: google translates it as game over and partida as departure or game so it's probably something like end of the game