"I like the remainder of the groups" is not prima facie good English. What is the context?

In the sense that you're using it, they are exactly the same (including formality). "The remainder" is what's left over, "the rest" is including everything all together that wasn't accounted for....so the same working definition. Remainder doesn't really have any other working definitions; however, "rest" does and I would be careful using it when appropriate.