I had the same sense of one connotation of “instantaneously” (”over almost as soon as it has begun”) as “Jeepien” did, although I still see an element of immediacy in both words.
To me, even though both “instantly” and “instantaneously” mean “right away,” instantaneously also carries some connotation of “for a short period of time.” Carrying that forward, I’d rather have the headache pill work instantly because, although it might be a stretch in this example, I wouldn’t want my relief from the headache to be short-lived.
I didn’t do any real research to try to validate my reaction, but I did do a quick Google search (search phrase: difference instantly instantaneously), and the first link that popped up was to a BBC World Service site called Learning English, in which people from around the world (South Korea, Tanzania, and India, on this one linked page alone, which addresses time phrases, adjectives and adverbs) write to clarify the subtleties of English (or at least British) usage.
The link is to:
http://www.bbc.co.uk/worldservice/learningenglish/youmeus/learnit/learnitv233.shtml
which says, in part: “If something happens instantly it happens immediately. If something happens instantaneously it also happens immediately but at the same time very quickly. Instantaneous and instantaneously are used only in a restricted range of contexts. . . [One Example] Death was instantaneous for all the people in the car when the bomb exploded.”
Even as explained by the BBC, this does seem to be an awfully “slice-the-salami-thinly” distinction (my apologies to Martha for not being able carry her frog hair analogy further), creating a nearly transparent slice of salami, but I thought it interesting that the first nuance that I (born and bred in the US) detected would prove to be the same as that of a BBC language service and of the first poster above, although I hadn’t seen either of those before I had the thought.
The more I try to think about length of time as a distinction between the two words, however, the more ephemeral it becomes, and I’m not sure I even find the BBC examples completely convincing of what I had started out thinking was my own point. Besides, as those who discussed Katz’s Deli earlier in the podcast know, there’s nothing less satisfying than an ephemeral salami sandwich.