Neanderthal and Homo sapiens have a common ancestor, that most people think of as Homo heidelbergensis.  However, since Neanderthal and Homo sapiens interbred, so anor way to think of this is to simply say that we are both just a sub-species of Homo heidelbergensis.  The ancestor of Homo heidelbergensis is some sort of later stage Homo ergaster. So, in anor way we can tell that Neanderthal man is direct ancestor of Homo sapiens.

According to microbial experiments, pre-existing helpful of advantageous mutations results in formation of new phenotypes. This finally results in a process called Speciation. Mutation also has a relation with creating new phenotypes.