I believe Enkidu also became much more wild over time. In Against the Grain the author suggests earlier tellings of Gilgamesh presented Enkidu as unusual, but human. When the same story was being recorded a thousand years later, he was a monster. If the story was preserving some ancient memory about Neanderthals, such significant change seems unlikely.
The epic was written around 2000 BCE which was well over >> 10k years after agriculture and more than 4 times that much after neanderthals died out.
It's possible but much more likely that they refer to something more contemporary. There are always "wild men" around no matter who you ask.