There are lots of pairs of words that mean the same thing, but one has a precise meaning and the other has a range of meanings. Now, if there are two words with subtly different meanings, I’m all for keeping both of them and retaining the richness of our wonderful language, etc. But when one just covers a subset of the other’s meanings, I’m not, to be honest, quite sure. At least here there seems to be a technical term lurking around which will keep the smaller (yet longer!) word going.
So, instantaneous, to cover the smaller meaning first, means being done or happening instantly. It does have a specific meaning in physics around being measured or existing at a particular time.
Instant means occurring immediately, as you would expect, as well as a precise moment in time or a very short time. It also means something that’s processed to allow it to be prepared quickly, in the case of food, mainly, Also, and I dimly remember this from when I learned to type in the Dark Ages, it means “of the current month” (your letter of the 16th instant) although surely no one uses that now?
Both of them come from the same original source, from Latin for “be at hand” (instare), but instantaneous came through medieval Latin, which added -aneus to the original instant (thank you, Oxford English Dictionary for that information). I would advise using instant unless you’re a physicist, just to save complication and make it easier to read.