You don't understand human natural language.

Chimps can communicate with us just fine and we can communicate with them. You have to be careful how you define it. We actually do not know whether Neanderthal, for example, had fully developed vocal apparatus but it is almost certain it did not. And it is not the vocal apparatus that drove human natural language, in all probability, it was the occurrence of things we can communicate that chimps can't (except by demonstration ... e.g., taking you to a place it hid something but vastly more than just this).

If Neanderthal's never interbred with humans then how come we have their DNA?

It is common for people to not examine what humans can communicate that no other animal can (except in terms of such things as demonstration). It is NOT that chimps do not think exactly like we do and may well be just as intelligent ... if they only had our organ for natural language (including improvements in neocortical wiring).

If you get into all the things we can express among ourselves with natural language, it isn't one or two things about what I brains compute for survival. As a major example, PRESENT IN EVERY SENTENCE, is tense. As another for 99% of all the verbs in any language, it is ambuity in agency (the communication asserts agency or does not -- always a choice unless said explicitly). Agency is blame. It kills people. ("That troll killed my child! Kill them all!).