Finishing a game:
Beating the final boss in a conventional game. Some games dont have a final boss, but have a clear objective. Some games have no "end" as well. It could also be unlocking all the characters and stages, in a game like Smash Brothers or Mario Party.
For example, in Mario 64 it would be beating the final bowser.
Completing a game:
This to me doing all the sidequests and post-game content. For example, in Mario 64 it would be to collect all the power stars the game has to offer. This can also be 100%ing the game in most cases.
In other games like Zelda OOT, it would be to collect all the gold skulltullas, obtain all optional items (like big gorons sword), and all the masks. You can also include the item upgrades like the quiver and bomb upgrades.
Personally, there are very few, I mean very few, games I have completed or attempt to complete nowadays. Some games take a lot of dedication, multiple playthroughs, mastering certain mini-games, and fully exploring the game world. I struggle to even "finish" most games I play, but it is at least the main goal for me with each game I attempt to start.
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