Homo sapiens apparently evolved in Africa, and spread out of Africa, populating the world. In the Levant, they encountered, and bred with, Neandertals who had moved south from Europe. We eventually either absorbed or replaced all other Hominids in the world, and became the sole Hominid left standing.
At first, archaic Homo sapiens shared the same stone tool technology with Neandertals, that is, Mousterian. In Europe, however, there was a sudden and punctuated burst in culture for the "Cro-magnon" group of Homo sapiens. Material culture flowered, microlithic tools were created and perhaps the ability to abstract came to the fore. The cave paintings of Lascaux are an example of this progressive flowering of the abilities of Homo sapiens.
Update: What it means to be "archaic Homo sapiens", and "modern Homo sapiens" is being modified by the science of genetics.