It sounds to me like you have one of two problems:
The steering wheel "button" for using the horn has died
or
The fuse carrying the signal from the steering wheel to the horn is dead, and since the alarm is routed to the horn differently it still works.
HOWEVER
Before ever tinkering with a steering wheel, ensure 100% that there is NO DEPLOYABLE AIRBAG. If you start to remove the steering wheel plate without deactivating this feature will return some explosive results.
I would first check the fuse for the horn, which would most likely be accessed somewhere near the dashboard, probably near the steering wheel or somewhere under the driver side dashboard.
Good luck =)