The real paleo diet was what they could get

The residue of what Neanderthals ate formed plaque on their teeth (as it does on ours).
DNA from the plaque is a snapshot of their diet.

Researchers are analyzing the plaque from Neanderthas' teeth to determine what they ate. What they have learned is that the genuine paleo people did not eat what modern diet fads call a "paleo" diet.

The Neanderthals ate pretty much whatever they could get their hands (and teeth) on.

By harvesting and sequencing that [plaque's] DNA, Weyrich has shown that there was no such thing as a typical Neanderthal diet. One individual from Spy cave in Belgium mostly ate meat like woolly rhinoceros and wild sheep, as well as some edible mushrooms. But two individuals who lived in El Sidrón cave in Spain seemed to be entirely vegetarian. The team couldn’t find any traces of meat in their diet, which consisted of mushrooms, pine nuts, tree bark, and moss. The Belgian Neanderthals hunted; the Spanish ones foraged.

“When people talk about the Paleo diet, that’s not paleo, that’s just non-carb,” Weyrich says. “The true paleo diet is eating whatever’s out there in the environment.”

But wait! There's more! They even had proto-pharmacies!

One of the El Sidron Neanderthals even seemed to be self-medicating with edible plants. One of his teeth had an abscess, and his plaque contained a parasite that causes diarrhea. But the plaque also contained Penicillium, the mould that produces the antibiotic penicillin, and poplar bark, a natural source of the aspirin-like painkiller, salicylic acid. The Neanderthal’s medical history—both diseases and treatments—were written in his plaque.

I guess they didn't brush their teeth.

And the Neanderthals didn't really go extinct - they just merged.