Take Tylenol not Aspirin (unless you have osteoporosis, in which take fish oil and glucosamine)

Aspirin (acetylsalicylic acid) is a carboxylic acid derivative while Tylenol (acetaminophen) is an amide derivative.(1p478, 506) Both act as analgesics for relieving pain and reducing fever.

By inhibiting the enzyme cyclooxygenase in prostaglandin synthesis, aspirin reduces inflammation making it a nonsteroidal anti-inflammatory drug.(1p478) In the stomach environment, the carboxylic acid derivative reacts directly with the stomach lining and can potentially cause bleeding.(2 & 3)

Tylenol acts on nerve endings to suppress pain, but is not a nonsteroidal anti-inflammatory drug.(3 & 1p506) In the stomach environment, because of its structure as an amide derivative, it doesn't react with the stomach lining and causes no bleeding.(2)

References

1. Denniston, KJ, Topping, JJ & Caret, RL. General, Organic, And Biochemistry, 5th ed. New York: McGraw Hill; 2007.
2. Kleiner, K. Fatty molecules make aspirin easier to stomach. Available at: http://www.newscientist.com/article/mg14519652.700-fatty-molecules-make-aspirin-easier-to-stomach.html. Accessed on Oct. 18, 2008.
3. Papazian, R. OTC options: Pain pain go away. FDA Consumer. Available at: http://www.fda.gov/bbs/topics/CONSUMER/CON00296a.html. Accessed on Oct. 18, 2008.