They inhabit a very special place in the natural order and it is likely they will always be with us from now on.
The Neanderthal plucked them from their place in the wild and turned them into something quite remarkable over the course of a quarter million years.
If you have ever seen comparative studies of dogs and wolves raised in captivity, their pattern of development is analogous to that of humans and apes. At a very young age they seem quite similar in abilities and design. It is only as they grow that the neoteny that is the pivot point of their biological cycle begins to produce remarkable effects in real capacity. By the time they are fully adult they are differentiated in an extraordinary fashion.
If you look at the changes required in terms of organic modifications that would be required to genetically engineer dogs to be capable of doing much more complex tasks like driving and speaking, they actually involve less alteration than would be needed to make men into better men. This is one of the reasons I am certain that Neanderthals developed much more slowly than Homo Sapiens - it is the simplest way to produce a better result in the end - by delaying maturity.