The second is not really a stand-alone sentence, but it could be a dependent clause, seeing "which" as a relative pronoun. Sometimes, to give the effect of being a bit of an afterthought, it might be written as a separate sentence. E.g. "His company is moving him to New York. Which is what he wants anyway."
You can't similarly use "that" as a supplementary relative pronoun anyway, whether or not in a separate sentence, but it might be a demonstrative pronoun, for example, when pointing to something. In that case, it would be a proper sentence.