What is the purpose of the relay in this circuit?
The current drawn by the compressor will likely 'fry' (over heat) the horn button. The relay can reliably provide LOTS (30 or more amperes) of current. Also the amount of current the compressor will need for the first second to so will be VERY high until the motor is running. So the relay is quite necessary.

The horn button will work fine to run the compressor, but only for a few operations, then it will be ruined.