Song topic has little relevance here, so don't consider that a selling point on your 'right' to the name.
Honestly, what are the chances of your bands coming across each other? If it's highly unlikely you'll ever be in the same area then I wouldn't worry about it. If it is possible, however, then I'd research who has been around longer.
If they have been around longer, then I'd suggest you either change your name or ask them if they'd be willing to change. If they decline, however, then you might not be able to do anything if they've been working under that name longer than you - that said, ask your attorney just to be sure. If you've been around longer, then ask them if they'd be willing to change. If not, you can pay a hundred or so bucks to have a lawyer send a letter to them to cease use of the name, and that will likely stop them.
Side note - you should practice some tact when you contact them. I doubt you meant this when you wrote up your OP, but it definitely sounded like you had a slight chip on your shoulder in regards to the name. It very well could be their name, and you might just have to suck it up and change your name to something else.