"Finishing" a game is just that - finishing it. Reaching the end credits. For me, there are three levels of "completion"...
Finishing a game, i.e. playing it through from start to finish for the first time.
Platting a game, i.e. getting all trophies for the game.
And finally, playing the f*** out of a game...this involves of spending dozens, if not hundreds of hours playing the game after (or rather than) obtaining the platinum trophy. Typically this is for games that I aim to achieve 100% completion or games with good online components, e.g. Demon's Souls