I’d say they aren’t quite synonomous: more like different meanings, same value.
The natural frequency is the frequency the system will oscillate at if displaced from equilibrium and released. E.g., stretch a spring and let it go. The frequency it’s vibrating at is the natural frequency.
The resonant frequency is the frequency with which you can drive the system such that resonance occurs. (I.e., the frequency at which the system will gain the most energy from being driven.) E.g., shake a spring back and forth with your hand. The frequency at which you should do this to get the biggest oscillations is the resonant frequency.
Typically, these frequencies have the same value. (I can’t think of any case where they wouldn’t.) So, if the natural frequency of the system is 10 Hz, the resonant frequency is 10 Hz. But just because they have the same value doesn’t mean they have the same meaning. Natural frequency is the frequency the system tends to oscillate at on its own, whereas resonant frequency is the frequency you should drive the system at for maximum effect.