No, we don't, but here is what we do know:
1) No Neanderthal mtDNA lineage has been found to date among several thousands of Europeans.
2) No Neanderthal mtDNA lineage has been found to date in fossils of early modern Europeans.
3) Models of intra- and inter-specific craniofacial variation support a clear morphologocal distinction between Neanderthals and Cro-Magnon.
4) The considerably shortened period of Neanderthal somatic development points strongly toward a clear distinction between Neanderthals and Cro-Magnon.
5) The clear divergence and monophyly of Neanderthal mtDNA suggests a long separation of Neanderthal and Cro-Magnon female lineages, four times greater than that of the most recent common ancestor of human mtDNAs. IOW, the most recent common ancestor of Neandertal and modern human mtDNAs is indicated to have been roughly 550,000 years ago.
6) Intermediate hominid forms stand between their most recent common ancestor and each respective lineage of Neanderthal and of Cro-Magnon.
7) We find no evidence of the contribution of Neanderthals to modern human genetics in haplotype tree analyses and performed phylogeographic analyses.
8) Neanderthal fossils cease to occur in a region almost immediately following the advent of modern human fossils into that region.
9) We find no mixed communities of Neanderthal and Cro-Magnon.
10) We find no hybrid intermediate forms signifying Neanderthal/Cro-Magnon admixture.
11) Cro-Magnons were the first to produce arrowlike projectiles tipped with ivory and amber, while Neanderthal weapons were only wooden spears sometimes tipped with stone points.
12) Cro-Magnons also made figurines and created objects of bone, an ability the Neanderthals apparently largely lacked.
13) Cro-Magnons learned to build the first true houses; there is only one known instance where a Neanderthal appears to have built a crude dwelling, quite possibly by imitation of Cro-Magnons.
14) It is almost certain that the FOXP2 gene that is critical in modern human speech and language was absent in Neanderthal. Silent polymorphisms in the gene date the critical mutations in the FOXP2 gene to the last 100,000 years, long after the last common ancestor of Neanderthal and Cro-Magnon.