Straight forward question. Are both correct or is one better. "Both are not." "Neither is." Also, are they interchangeable or are there correct times to use one or the other?
Both are not.
Both phrases are correct and furthermore, they have the exact same literal meaning—(!A && !B), or "not A and not B" for non-computer nerds.
As for when one might be preferable over the other, I actually can't think of too many differences between the two in terms of connotation. "Both" perhaps emphasizes the fact that there are two items of consideration, while "neither" perhaps emphasizes the absence of correctness. Consider
"Do you have any more candidates? You've sent me two and both are not qualified."
and "Neither is satisfactory and we are out of options."