The only place on earth I could see another species theoretically holding on is Iceland. Let's say the Neanderthals manage to sail there, somehow, in the late ice age. Their population would likely be tiny, in the range of a few thousand, due to the lack of food besides from the sea (Tasmania only had 5,000 prior to contact for comparison). They would not have been on Iceland for long enough to dwarf or change substantially from their ancestors.
Prior to the viking arrival, Irish monks come as they would have otherwise, but form a more permenant base in an attempt to christianize the Neanderthals.
Let's just say, for the sake of argument, Neanderthals did have minds that worked roughly like ours, and complex language. Christianizing works to a certain extent, with a subset of the population converting, and most everyone gaining the benefit of new technologies to a certain extent.
Vikings arrive. Irish monks manage to slip out almost all of the Christian Neanderthals. They become established as a community in Ireland to this day, becoming fully integrated, albeit losing almost all of their culture in the process.
The ones left behind are, in the beginning, almost unmolested by the Vikings. They have nothing worth plundering. They also neither use the interior of Iceland (no native land animals), or go far into the ocean, so they do not initally cause much friction with the pastoral culture that develops. Considering the Vikings have a reputation as warlike, this may seem odd, but they had experience with stone-age finns and lapps and similarly did not slaughter them needlessly.
However, the superior strength of the Neanderthals becomes obvious, and they're soon hired for odd jobs. Like most pre-agricultural people, they do not take well, culturally speaking, to long, tedious work. As conflicts begin to arise with livestock raiding, groups are dispersed, with adult males killed, and women and children brought into bondage. The population just barely survives, because they cannot breed true with other humans, and once acculturated, the children submit to slavery...about as well as normal people do, meaning enough to suffer, yet still reproduce.
Neanderthal population undergoes an explosion by the 1300's due to drops in infant mortality. There is no plantation style system, but many small landowners have slaves for the heavier field work, barn raising, etc. Surplus slaves filter into Norway, and eventually Denmark, though not into Sweden. A minor slave revolt results in a group escaping to Greenland (where their decendents were rediscovered in the mid 1800s).
Some minor historical butterflies hit the U.S. upon founding. A scattered few Irish Neanderthals immigrate, and seek to claim rights as men, but immigrants from Norway to the upper midwest claim that they are allowed slaves on free soil because Neanderthals are not of the race of men. The supreme court om the 1850's narrowly rules that slavery is slavery, and uses the testimony from some learned Neanderthals from Ireland to suggest that "the level of civilization that can be gained by these men is superior to that of the negro". Many Norse immigrants transplant to Oklahoma as result.
Slavery is outlawed in Denmark proper in 1876, and Norway in 1905. Iceland holds on through the 1940s, but one Irish MP (of Neanderthal decent) makes an impassioned speech at the United Nations resulting in a condemnation of Iceland. Given slavery at this point was nomial economically, the remaining slaves are released.
Thoughts? This is rough of course.