There seem to be two Latin adjectives that mean "Asian": Asianus and Asiaticus. The dictionary entries in Lewis and Short linked above suggest that the two adjectives are different, but no comparison is given. What is the difference between these two words? When should I choose one over the other? Are there cases where both are equally valid and mean the same thing? Are there cases where both are possible but the meaning is different?
There's actually another: Asius. There aren't many differences between the three, and they could all roughly stand for something that we would call "Asian" (keep in mind the differences between modernity's Asia and Rome's Asia). Asius and Asiaticus come from Greek while Asianus is Latin, and thus typically is most often used when referring to the province of Asia Minor. For things like "Asiatic oratory," both are common (in reference to the style, Asiaticus is more common, but Asianus isn't rare).
Asius has a couple special uses:
B. Adj.: Ā^sĭus , a, um, of Asia: “palus,” the marshy region on the river Cayster, Verg. A. 7, 701; cf. id. G. 1, 383, and Hom. Il. 2, 461; “Asia,” a nymph, Verg. G. 4, 343; cf. Hyg. Fab. prooem.—
You will also see Asiani referring to people far more often than Asiatici, and in fact, I'm not so sure you'll see the latter to refer to people from Asia (or Asia Minor).