Excedrin and other APC preparations have only one benefit over plain aspirin swallowed with coffee: for some people, there is a beneficial placebo effect with certain brand names. More power to them.
However to some extent the placebo effect is dependent on ignorance that you have been given a placebo, so if you used to find that Excedrin worked better than aspirin with coffee, I may have just destroyed that benefit for you. Too bad. Find another sugar pill.
Acetaminophen (paracetamol, Tyelnol) is no more effective than aspirin (acetylsalicilic acid or ASA) in pain relief, and IIRC, does not potentiate with ASA nor are the two taken together any better than either one taken alone when taken in the usual recommended adult doses. But I'm pulling that from 30 year old memories from my days as an RN so someone could check on that.
There are downsides to both acetaminophen and aspirin.
Too much acetaminophen can irreversibly damage your liver, possibly leading to death or a liver transplant. Since the stuff is indiscriminately added to a huge number of pain relief compounds, it is possible to OD without realizing it, especially for those who think that since the Excedrin they took 15 minutes ago isn't taking care of the headache, they'll just take some Tylenol now, and maybe wash it down with Alka Seltzer Plus. Bye bye, sucker! (BTW, if you are contemplating suicide, don't do it with a couple of bottles of Tylenol. For if the paramedics get to you in time to pump your stomach and save you, you will live the rest of your short miserable life without a working liver, and that is hell on you, and everyone who has to breathe the stenches you emit.)
Too much aspirin can permanently damage your hearing, or kill you in a number of different ways, or turn you into a semi vegetable through bleeding into your brain. One thing is that an early sign of a mild OD is tinnitus, which is a high pitched whine from damage to your inner ear. It is generally reversible by abstaining from more ASA for a time. So there is that warning sign for some people (but maybe not for you and maybe not all the time so don't rely on it).
Aspirin washed down with a cup of coffee is as effective as any of the fancy brand name compounds in treating migraine and the aches and pains of daily living. Caffeine definitely potentiates the analgesic effect of aspirin, while also acting directly on the blood vessels that cause migraine pain. Plain coffee is the preferred way of delivering the caffeine, since it contains a number of other drugs that also have some benefit, and because a warm solution more quickly gets into the blood stream than a cold one laden with sugar, etc. This statement is true when the possible benefits of a placebo reaction to a given brand name drug are discounted--- but we have already destroyed that placebo effect for you with this post, eh?
Bayer aspirin is generally priced twice as much, or more, than just plain aspirin, but is no different from any of the rest once it is out of the bottle. Still if you want to enhance your pain relief with a bit of placebo effect, Bayer might well work better for you than the cheap generic stuff.