Well, off the top of my head, I'd say a lot of things. (In the eyes of the law it'd be the difference between Destruction of Property and Murder, for one thing.) But that's my point, that it would be better to describe what actions would cause the spell to fail, rather than a list of circumstances in which some actions may or may not cause the spell to fail.
If you use an axe to cut a rope, you remain invisible. But if you swing the axe and hit the invisible goblin sitting on top of the rope (who is there to make sure no one cuts it, but doesn't see you as you are invisible), and you accidentally hit him (rolling high enough to hit his AC, and the overcoming the 50% miss chance), you become visible. Its the same swing of an axe, but the end result is radically different.
So, what is the difference between swinging an axe at a goblin, and swinging an axe at a rope?