Since it works with an extension cord, it seems like you do not have 120 volt power to the outlet where the water heater is plugged in. You should have a 20 amp circuit breaker in your electrical panel which you say you have checked.
To go any farther you need to have access to an electrical meter or you can just change the circuit breaker in the electrical panel. In my Monaco coach, the water heater is on it's own circuit, without any GFI breaker, etc. Since it is the only item in the circuit, either the breaker is bad and provides no power when it is reset, OR a wire has come loose somewhere between the circuit breaker and the power outlet.
If you are not comfortable working with electric it may be time to ask a friend that has a voltmeter to locate the problem. Circuit breakers do occasionally go bad and the only way to be sure is to reset it check the voltage out.