Michael's post (and the similar one's) hit closest to center-target.
Acetaminophen/paracetamol technically isn't an anti-inflammatory, and that's why ibuprofen generally works better on pain from pulled muscles, sprained ankles, etc. It does work really well on headaches, and is a great antipyretic. Thus, it works well for random viral illnesses or 'flu' that cause fevers, aches, and pains. Also, ibuprofen is notorious for upsetting stomachs, so it is best taken with food.
The good news is that since they aren't in the same class of drugs you can take full doses of both at the same time- as several people have said.
As an aside- another BIG reason that acetaminophen causes so much liver failure in the US is that people think it is pretty safe, and thus depressed people who try to get attention by making a suicidal gesture (but who aren't REALLY trying to kill themselves) will overdose on Tylenol. Little do they know, but they just killed their liver and now need a transplant- IF a liver is available for them. If not, liver failure is an exceedingly undignified way to die...