If his team gets ahead, he'll generally ensure they stay ahead. If his team's behind, he can turn the situation around in a team fight, but that's not guaranteed. He's generally quite good with a competent team, so at high ELO, he can be downright scary, but at low ELO, teams that don't make adequate use of his enormous team fight presence can lose when they try to play assassin style. Also, he tends to be a little lacking in the 1v1 department late game so your game play as a very fed Amumu would be different from your game play as say, a very fed Jax.
Tldr: I've seen people catch up thanks to Amumu. I've seen people maintain their lead thanks to Amumu. I rarely see teams that initially win but then lose with Amumu unless it's one of those 50+ minute games where ADCs on either team take people out in 2 seconds. He's very strong with a competent team, especially an AOE team
I use to main him in season 2, before he became popular and got nerfed and even then, I played him often. I'd drop about 10 pink wards a game and still have a huge impact in team fights.