I am not a paleontologist, and I hate to disagree with one, so I will let other paleontologists do so instead. In post #211 I linked three relevant journal articles. One of these goes directly to what I suspect your teacher alluded to:
The Neanderthal taxonomic position: models of intra- and inter-specific craniofacial variationThe morphological distances between Neanderthals and modern humans, and between Neanderthals and Late Paleolithic/early anatomically modern specimens, are consistently greater than the distances among recent human populations, and greater than the distances between the two chimpanzee species. Furthermore, no strong morphological similarities were found between Neanderthals and Late Paleolithic Europeans. This study does not find evidence for Neanderthal contribution to the evolution of modern Europeans. Results are consistent with the recognition of Neanderthals as a distinct species. [bold print added]
The second, although I don't think it's precisely what your teacher had in mind, is a difference between modern humans and Neanderthals that is without question greater than the difference between any two modern humans: Surprisingly rapid growth in Neanderthals. What this study finds is that Neanderthals reached full maturity (adulthood) by about age 15. No modern human reaches full maturity by age 15.
Finally, there is the implicit distinction of the FOXP2 allele that exercises a crucial function in human speech and language. This mutation is dated to hundreds of thousands of years after mtDNA dates the last common ancestor between Neanderthals and modern humans. Every modern human relies on this mutation to speak properly; it is implicit that this mutation was absent among Neanderthals.
But, as has been noted over and over, there is sharp disagreement in paleoanthropology circles about all of this.