Without stressing any particular word in either sentence (which stress can add a meaning of its own), the constructions allow for -- but do not necessarily have -- the following in subtlety: 'John hasn't married yet.' => This sentence uses the present perfect, therefore it has some psychological relevance to the speaker at the moment of speaking. Perhaps the speaker leaves open the chance that John will marry in the future. John hasn't married yet, but there is a good chance that he will someday. On the other hand, it could mean: John hasn't married yet and he never will. But, overall, despite these possible opposite meanings (and there are many more possible meanings that can be intended by the speaker due to the use of the present perfect), the sentence does not seem abrupt as can the second sentence. 'John has yet to marry.' => This just seems a statement of fact. The sentence is a little abrupt. The sentence *can* (but does not have to) mean that the speaker thinks that John should already be married. Of course, overall, it lacks that "special something" that the present perfect brings with it: some psychological relevance to the speaker at the moment of speaking, and that the speaker might expound on. If I wanted to be abrupt or matter-of-fact I would choose this sentence. If I wanted to talk about possibilities, I would use the first.

The words are the same. But each could have very different tone when spoken and perhaps other non-verbal communication such as raised eyebrows. You could use each to suggest John was getting married very soon, was gay, had a pregnant girlfriend or any number of other suggestions.

they mean the same, although we would probably say John isn't married yet John is yet to marry

On a simple note. The idea is the same. John is not married.