Both seem to mean with the support or protection of. Are they interchangeable? Mostly interchangeable? Is there a difference in connotation?
For the most part these are synonyms, although I think auspices carries less a connotation of protection and more of patronage or sponsorship, while aegis has the opposite feeling. WikiDiff and similar sites are a good resource for exploring this sort of subtlety.
They are similar in meaning, yet have different origins - one Greek, the other Latin.
Aegis is from the ‘shield of Zeus’ and means you are under someone’s wing, under their protection, or ‘shield’. (See below).
aegis (n.) "protection," 1793, a figurative use of Latin aegis, from Greek Aigis, the name of the shield of Zeus, a word said by Herodotus to be related to aix (genitive aigos) "goat," from PIE *aig- "goat" (source also of Sanskrit ajah, Lithuanian ožys "he-goat"), as the shield was of goatskin. Athene's aigis was a short goat-skin cloak, set with a gorgon's head and fringed with snakes. The exact use and purpose of it is not now clear.
Auspices has an origin which surprised me - it originates from ‘observing birds to take omens or predict signs of the future’ and extended to become ‘protected by’ or ‘under the care of’. (See below).
auspices (n.) plural (and now the usual form) of auspice (1530s), "observation of birds for the purpose of taking omens," from French auspice (14c.), from Latin auspicum "divination from the flight of birds; function of an auspex" (q.v.). Meaning "any indication of the future (especially favorable)" is from 1650s; earlier (1630s) in extended sense of "benevolent influence of greater power, influence exerted on behalf of someone or something," originally in expression under the auspices of.
When wanting to understand the real meaning of words, rather than relying on searching for common usage online, I suggest looking up the etymology, which gives you the original source meaning.