It means that whatever it is, is in the past. We can't do anything about it and it should be forgotten or put aside. Could you catch water that had gone past a bridge that you were on? It certainly wouldn't be worth the trouble. "I wanted to apologize for the argument we had last week." "Don't worry about it. That's just water under the bridge."

As Fdmaxey said: Let bygones be bygones. But also: A lot of water has passed under the bridge since you left: Many things have changed since then.

Something that happened in the past and cannot be changed. For example, It's no good to regret what is under the bridge. Also, water over the dam has the same meaning.