There isn't a real difference there. The only technical difference would be that "Ich suche mir eine neue Stelle" would eliminate the option that you are looking for someone else because of the "mir". But usually, if you'd say "Ich suche eine neue Stelle" everyone would assume that you are talking about yourself. But just to be technically correct "Ich suche eine neue Stelle" doesn't imply for whom you are searching. But again that is really just a technical difference to be really exact.