If you read the "Hypotheses" paper that was delivered in Poznan in 2009, you'll note that there are several criticisms of OT along with the areas of overlap between the theories. I personally think that the biggest problem with OT is that it still adheres to basic generative assumption that grammars exist primarily to validate structural well-formedness. However, it does share one very radical notion with Natural Phonology (NatP) that did not exist in classical generative theory--the lack of any constraints on inputs, so-called "richness of the base".
From a NatP perspective, I would just put it this way--people can try to articulate any phonetic target they can imagine. That is, one can try to say "bnick", even though the initial cluster is clearly ill-formed for English. NatP proposes that there are two basic types of phonological processes: fortitions and lenitions. Fortitions (like CON?) are clarifying processes that tend to make different sounds more salient and less like each other. So epenthesis would be a fortition. Lenitions (like EVAL?) are simplifying processes that make sounds more like each other in context (e.g. assimilation) and tend to reduce structure. So syncope would be a lenition in that it reduces the number of syllables produced during execution. Unlike OT, however, NatP allows easily for processes to produce rather opaque forms. So voicing assimilation can be "fed" by syncope, and voice assimilation might "feed" other lenitions. So you can get extremely radical deviations from input in fast and casual speech. (Stampe once wrote a short paper called "Divinity Fudge" published by the Chicago Linguistic Society that went into detail on the wide range of casual pronunciations of that phrase.)
The theory of Natural Phonology holds that infants start out with roughly adult phonemic forms even when the child produces extremely deviant outputs in relation to the underlying forms. Hence, NatP is able to explain the stages that children go through as their process system is gradually brought under control. OT has nothing much to say about speech outputs in the earliest stages of language production.