I think the main difference between the two is the *amount* of regret. I think, and this is just my thoughts mind you, that regret is just the basic feeling at a surface level while remorse is a more in depth form if regret. The two are essentially the same idea and you can make the same reference with both words, but if you want more emphasis, go for remorse. If you want to just highlight the emotion, go for regret.
