TL;DR; Lord Voldemort abilities and powers remain the same, phisical aspect remains mostly the same by the choice of magic used.
First. We need to assume in the HP universe there's such a thing as a soul and that soul, and not the body, is the source of life and sentience.
In that regard (as it is in many religions, old and very old) the physical form is just a mortal embodiment for the immortal soul as we go trough this life.
As such, and specially as hinted by the archway gate at the departments of secrets on the Ministry of Magic, once the mortal body ceases to be, the soul continues its journey somewhere else which is not known to us. If the soul refuses to go on it stays as a ghost without a material body or attachment.
The creation of a horcrux involves two steps:
- Splitting your soul
- Attaching the extra piece of soul to a material object and magically bonding it with the material world.
Now, there's an assumption (shared with many religions) that a soul is not only immortal but also infinite, meaning splitting your soul in two will still let you have two whole infinite souls. But now both of them would be tied to the material world so, if the material embodyment is no more, your sould does not move on but stays trapped in the horcrux with limited ability to interact with the world but with the possibility of inhabiting a new "host".
Thus, to go back to your question, Voldemort after being resurrected was brought back to a magically created body which was as closed as he could get to his original body. Some things were changed, some due to the use of dark magic to recreate the body, but the essence of his appearance was the same (he was still recognized by the ministry officials). It is never explained why but one is to assume the magic required to get the soul back to the body is limited and can't get you back to just any body. If diary tom would have succeded he would have come back as a young tom... So we must assume is the magic creating a body that definws the appearance.
Regarding his powers and abilities the same applies, his body can now resist harrys touch, but that is just magic to make his body resistant. All the rest remains the same as the horcrux just allowed his sould to remain on this world and to, eventually, go back to a phisical body.
As a side note, the damage to the soul comes not from the actual splitting but from the evil act that is required to split the soul. The act is so evil the soul itself is split in two, and after doing it seven times, the soul was so corrupted and deformed that it has the form we see at the end of book 7.
Defining whether he is human or not is difficult. If you go with a definition based on his soul he may not be, if your definition is purely phisical then he is, or at least he is as human as Moody is, who has half of his body replaced or enhanced by magic.