The ice age that started up when the humans started to move in on the neanderthals would still be in effect, and if the asteroid is used as the POD it is much more severe. They would migrate south, keeping with the low tundra habitat. Megafauna makes a major comback as the woodlands die out, so Mammoths and the like are still around for a while. I'd see a major bottleneck as the neanderthals move south, they were already at roughly 100,000 individuals, so they'd probably drop to 500-10,000-50,000.
With the expanded glaciation, they would eventually make a rebound, and reach a larger population than pre-glaciation levels. They would remain at status quo for a couple millenia, until the glaciers recede. At that point they would have to do what our simian ancestors did, run from habitat to habitat as the niche dies out. Instead of island jungles in savanah, it would be island taiga in steppe. Most will go north, but a few will adapt to the warmer conditions and become taller, less robust, faster, etc.. Resembling modern humans, but with different facial structure and size.
The neander-humans would be much stronger than us, slower, shorter, stockier, but less so than original neanderthals, imagine a race where the average is an NFL linebacker.
Culturally they would be similar, but be far more stationary than us, a remnant from their roots. They'd move either when it is absolutely necessary, or to follow a food source, like Native Americans and bison. They would be a bit more peaceful than us, again because they are immobile so they would keep the same power structure to keep the peace, whereas humans have fluid leadership from the nearly constant moving and hunting. Their intellegence would be a bit lower than us on average, but close enough so they have many individuals who rival our own gifted people. Their Einstien would be our Bill Gates.
They would develope a bit slower than us, but would basically follow our path. When nations are developed war would be much less common, used after diplomacy fails, but it would be far more brutal, due to their sheer physical strength, this may be factored out when long-range warfare becomes the standard, around the Renaissance for us.
Of course, the various adaptations may change their mental functions or ease of anger but I don't want to go insane.