This is an interesting question, and a complicated one.
Why Neanderthals Died Off
First off, let's examine some of the reasons why experts believe neanderthals died off:
This species had incredible muscle mass, and matured earlier than us. Physically, they were much stronger than to us - a neanderthal in his prime may have been able to rip an arm off of a homo sapiens and beat him to death with it.
However, their great strength was also their biggest weakness: they could not scavenge the supplies needed to keep themselves, let alone their pregnant women, and children alive.
Basically, it is believed that they starved to death, as a species. They were also competing with homo sapiens, who required less food, and were, apparently, just a little bit smarted when it came to using tools and adapting to new situations.
Meeting Each Other
Our ancestors clearly spread throughout the entire world. These tribes traveled, by foot, across continent and oceans (over ice bridges, etc.). However, with the world frozen over to a large degree, the routes they would have followed are relatively limited. These species would have met sooner rather than later.
Being tribal in nature, and basically savages, there's only two ways their encounter could go: war, or inter-breeding.
Either way, fast forward a thousand years, and you're left with pretty much a single species.
Your Specific Question
But let's say that some Neanderthals did not starve out, and met up with homo sapiens only a few thousand years later. They both posses similar levels of technology.
If the species can't interbreed, then they are simply competitors for the same resources. Until very recently genocide was a very acceptable answer to dealing with your competition (native Americans, anyone? What about the Mayans, or the Conquistadors? When other people have stuff we want, we've killed them off mercilessly, and we're the same species).
One species would kill the other off, with my money being placed on homo sapiens - our ancestors were simply smarter.