How was it possible for Neanderthal Man and Homo Sapiens to evolve as separate species?

Geographic isolation and different living conditions.

Explanation:

Both the Neanderthal and the Homo sapiens shared the same ancestor, the Homo erectus. The Homo erectus managed to spread from Africa to Eurasia. The populations started to become geographically isolated from each other, and over time this caused speciation.

The Homo erectus from the far East, with the one from Europe, or the one from Africa, didn't had contact. This was due to big distance and natural barriers. Gradually this species was evolving, and because of the different living conditions and the geographic isolation at different places it started to develop different characteristics. This has led to evolution of different species over time, with the Neanderthal evolving in Eurasia, as well as the Denisovan Man, while in Africa it was the Homo sapiens that emerged.