Plum curculio, bacterial spot, brown rot, oriental fruit moth, and possibly even the codling moth are the most significant problems apricots face, but the first three are primarily found in the eastern half of the United States where the climate is fairly wet.  Brown rot is also responsible for twig and blossom blight, and while the latter can be just as devastating on fruit production for some trees, I am not sure if it is a common problem on newer varieties due to how difficult it is to find information about it.  Overall, apricots can require high effort to manage, but this doesn't even account for their early blooming habit, which makes them exceedingly vulnerable to frost damage in most of the country. 

Apricots regularly experience sudden death in wet environments shortly after the leaves sprout.  The reason for this isn't entirely known, but the citation rootstock, and possibly St Julian A, may be involved (when combined with this species), both of which are often used.