2003 acura tl horn does not work when pressed but alarm horn works

The horn button could be bad, but that is a very expensive fix most likely. First thing to do is check ALL the wires you have access to, paying particular attention to the horn wires running around the floorboards and steering column. Chances are good the wires have been damaged. Considering the potential expense of replacing the steering wheel (I assume the button is on the airbag on the steering wheel) I'd suggest wiring a new switch somewhere on the dash for the horn if it turns out to be the switch, but I'm reasonably certain the problem is a damaged wire.

If the alarm horn is the same unit used as the car horn then your problem is the contacts for when the horn is pressed. Some horns are activated by pressing the large center area of the steering wheel others have little buttons on the left and right side of the steering wheel. If the latter is the case make sure you try both pressure buttons to see if horn will work. Either way the contacts have to be cleaned or the wheel needs to be adjusted to allow the contacts to touch each other when pressed. That's most likely your problem.