It would seem the fob is working as it is supposed to, but the horn is not.
There could be a problem with the horn system itself and that might need to be checked out. Does the horn work if you depress the horn pad /buttons on the steering wheel?

 

 

Usually,,,,,, the horn button /contacts make the ground to operate the horn relay which completes the path for power to the horn(s). If your vehicle has 2 horns, it is rare that both would give out at the same time, so you could check the horn relay out . If the problem is after the relay you should hear the relay click when the horn pad is depressed. If you don't hear it click/work , relay itself is suspect. Check with fuse/relay section of your owner's manual and it will show which relay controls the horns. Once you see it you might find another relay in the panel that has the same # and to prove out the relay just switch the relays. I would not leave the relays switched because the good one may serve a more critical function. Then all you need to do is get a new one.