If the **** sapiens weren't purposely competing with the Neanderthals, it doesn't mean they weren't doing so beyond any willfull intent. The early humans would've just moved in slowly and pushed the Neanderthals back without malice.
Early humans did live among the Neanderthals since 130,000 ya, but that was mainly in the Middle East area. When Cro-Magnon and other early humans pushed into Europe in the warmer point between the two Ice Ages (c. 60,000-20,000), they essentially got a toe-hold there. Then when the last Ice Age hit the competition was too intense for the Neanderthals to survive. The Neanderthals couldn't compete. For instance, they weren't
even anatomically capable of throwing a spear. Nor were they capable of using bow and arrows. The **** sapiens just out-hunted them forcing the Neanderthals into more delicate locations.
I do think it's possible that the Neanderthals could've survived in the Arctic up until modern times, if the Ice Age didn't happen.