Ben, I'll try to answer that question; it's not personal at all, those three drugs all have different effects, warnings, interactions, and symptoms on which they work best. Keep in mind my medical education comes from books and wikipidia (@_@)!.. Anyway, asprin is good for fever, mild pain, inflamation, and thinning the blood. Ibuprofin is good for the first three only. Acetaminophen is good for fever and severe pain, but not the other issues. The first two are NSAIDs, which can damage stomach linings, and can be hazardous for asmatics or people with stomach ulcers, or open bleeding wounds, etc.
Acetaminophen on the other hand can be bad for the liver, so I wouldn't offer it to someone with partial liver failure (for example last Saturday night on a company sponsored drinking binge, my co-worker suffers from partial liver failure, but had a splitting headache.. so gave him the other two).
Also, carrying both types allows me to treat more severe pain without overdosing on NSAIDs, by giving one dose each of NSAID and one non-NSAID. Thats the combo I'd take to get back to the trail head with a sprain/fracture, god forbid..
Oh, just remembered, Ibu. is the best of the three for dysmenorrhea (womens cramping), and my GF occasionally wants one for that.. I could go on and on.. no ibuprofin for pregnant women, no asprin for bleeders, etc.. you get the idea.
Funny thing is; I rarely need any of this stuff for myself!