Beyond the particulars of Neandertals and modern humans, though, X-Men’s abundant speeches about the "next step in human evolution" only make sense if evolution is seen as the gradual realization of some design embedded in nature. Accordingly, the emergence of superior new species—not just new species in general but particular, better species—is supposed to happen, and woe betide the older ones that stand in the way of that progress.
Current conceptions of evolution reject those views, however. Smart evolutionary biologists avoid referring to hierarchies of species in which, say, apes count as higher organisms and snakes as lower ones. Species are understood to emerge only if environmental conditions allow new, distinct breeding populations to branch away from their predecessors. The prospects for new species to survive or replace ancestral ones are equally hit or miss.
Mutants in the X-Men films are always treated as a distinct species, but most of them can apparently pass as human and spawn children with them. Those facts do not absolutely eliminate the possibility that the mutants are a different species (because related species do sometimes limitedly crossbreed in nature and bear fertile offspring, as wolves and coyotes have). Yet it also isn't clear that the mutants would preferentially breed among themselves, as a species must under at least one widely-known biological conception of a species.
Recognizable species also usually have a definable phenotype, or set of characteristic physical features. The X-Men mutants, in contrast, are a crazily diverse mélange of types (teleports! banshees! living ray guns!) who are on average at least as different from one another as they are from the rest of humanity.
So contrary to the film’s heroes and villains, X-Men mutants are not innately a new species, just another variant of Homo sapiens. They cannot become a new species (or more than one) unless geophysical or other circumstances create an irrevocable barrier to their breeding with the rest of humanity. Tolerance of one another’s differences could be enough to prevent that outcome. So in that sense, notwithstanding the dodgy science along the way, the film's underlying message is probably right after all.