That grammar evolves naturally, like species do, so any rules we find for the categorization of either should reflect reality, not try to dictate it.

For example, I just started that bit with a clause, which means it’s a fragment, not a sentence. I still put a period at the end of it and started it without any lead in, which is “wrong,” but it’s more that the rule is wrong, because what I wrote plays the role of a sentence in this case.