First, I'm going to state the obvious:
Drugs work by interfering with the body's natural functioning to create a desired effect (as well as many side effects, aka undesired effects). You cannot chemically manipulate a biochemical pathway without starting a cascade of unwanted effects.
Advil, as an NSAID (Non-Steroidal Anti-Inflammatory Drug), interferes with the biochemical pathway that encourages inflammation. However, this same inflammation is what triggers the anti-inflammatory cycle which is responsible for laying down new healthy tissue and repairing the area. If you have damaged tissue and take Ibuprofen (Advil), you are actually impeding your body's ability to heal.
Tylenol (acetaminophen), on the other hand, works with a different pathway, but that pathway is also responsible for T cell production. T cells are one of our basic immune cells and are used to target invading organisms and eliminate problem cells (bacteria, viruses... even cancer!). These are the same cells that are not being produced in those with AIDs which is why people with AIDs can be killed by something as seemingly harmless as mold spores. They no longer have the cells to defend themselves.
Many athletes talk about using Advil post workout, but workouts shred tissue. Post workout, your body is working hard to rebuild and repair... is it a good idea to impede this process? Research has shown that neither Advil nor Tylenol improve recovery time from a workout.
So if Advil impedes the healing process, and Tylenol impedes the immune system, which would YOU choose?