Neanderthals were thought to have died out as modern humans arrived in Europe. Now, artifacts found in a cave in Gibraltar reveal that the two groups coexisted for millenia before Neanderthals finally dwindled out of existence...

They did not share a cave, but rather shared Europe. There isn't any evidence of the two inter-breeding, so it looks they just lived in the same continent at the same time, perhaps peacefully.

My belief is that the cataclysm of around 10,000BC caused the demise of the Neanderthals, and the mutation of humans into how we are today, as well as other human mutations that did not survive until today - giants, hobbits etc. The Neanderthals and earlier humans are mutated forms of an earlier homo species, wihich occured during the previous cataclysm, whenever that was...

Source: New Scientist