New paper which is very similar, but subtly different, from previous paper
What I would do, assuming this new development is worthy of publishing (which I cannot guess without knowing the contents - and the reviewers will do anyway):

Write the paper (the new one), in a self contained manner. Your previous work is a related work, goes in the bibliographical review. Compare and position your new paper wrt the old one, in the same way you would do if it was from another author. Including using the same proofs as references ("Theorem 4 from [1]"), for the ones that do not have differences. Don't copy stuff...

There is no problem if you paper is similar to another one. Just highlight the similarities and differences. I usually like such works, because subtle differences can be derived from different interpretations of the same stuff, which can lead to very cool stuff :)