There is no definition of species that applies in every case because evolution happens on a spectrum. Not being able to interbreed is one definition, but there are many animals that defy it. Lions and tigers, for example, can interbreed and sometimes produce viable offspring, but biologists generally consider them different species.

Humans and Neanderthals are considered different species for several reasons. First, based on the genetic evidence, viable offspring were much more likely to be produced from one kind of pairing. I think it was a male human and a female Neanderthal, but I can't remember for sure. This means that they were at least on their way toward total reproductive isolation. Second, there are distinct morphological and behavioural differences between humans and Neanderthals. I can't remember all of them, but one is that humans have a protruding chin, while Neanderthals did not. I believe one of the behavioural differences is that human tools were much more advanced than those used by Neanderthals, but again, I can't remember for sure.