why DFA to regex by two different methods differ
No, these methods are not guaranteed to yield the shortest regexp.  Therefore, there's no reason to expect them to necessarily yield the same regexp.

Given a regexp $R$, finding the shortest regexp $R'$ that is equivalent to $R$ is known to be PSPACE-complete, by a result of Meyer Stockmeyer.  Consequently, none of these procedures can be expected to yield the shortest regexp.  If they did, then we could just take $R$, convert it to a NFA, then convert it back to a regexp $R'$ using that method, and we'd have an efficient way to find the shortest regexp, implying that P = PSPACE -- something that is not believed to be true.

(In fact, it's even hard to approximate, i.e., to find a regexp that is close to the shortest.)

To learn more, see On Learning Regular Languages on the CSTheory.SE blog.