Some mules are not sterile, though that is rare; most mules are. Neandertals and sapiens were isolated by only a few hundred thousand years, whereas horses and asses were isolated for millions of years. Same with the cat species.
There are isolated populations of frogs in the US in which each population close to the next produce viable offspring, but the most distantly separated populations cannot produce viable offspring when mated.
Neandertals were likely, therefore, actually within the same species as sapiens. There is some indication, however, that their ability to speak was very limited, compared to the more advanced sapiens ("strong, silent type"). Perhaps their occasional mating with the sapiens females led to their extinction! (Guys usually don't like strangers messing with their women, particularly if from a much different race. Such mating might well have led to tribal wars, with the more advanced sapiens prevailing.)
There are isolated populations of frogs in the US in which each population close to the next produce viable offspring, but the most distantly separated populations cannot produce viable offspring when mated.
Neandertals were likely, therefore, actually within the same species as sapiens. There is some indication, however, that their ability to speak was very limited, compared to the more advanced sapiens ("strong, silent type"). Perhaps their occasional mating with the sapiens females led to their extinction! (Guys usually don't like strangers messing with their women, particularly if from a much different race. Such mating might well have led to tribal wars, with the more advanced sapiens prevailing.)