Translation:No, my health does not permit it.
More or less. Technically they have similar meanings, but you should try to use one over the other depending on the situation.
Usually allow is generally used as a verb, such as in this case, although to be fair this whole sentence is not natural english. Permit, on the other hand, is most often used as a noun, e.g. "he's carrying his permit" or "I bought a parking permit, so I can park here". I can't think of a situation where I've heard someone use permit as a verb.