From my experience, a vast majority falsely believes FM solely modulates the frequency and keeps a constant phase angle while PM keeps a constant frequency and the phase angle is being modulated. Even Wikipedia  seems to imply this: “[…] In frequency modulation, the modulating signal causes the carrier frequency to vary. These variations are controlled by both the frequency and the amplitude of the modulating wave. In phase modulation, the phase of the carrier is controlled by the modulating waveform.[…]”
Keeping a constant frequency while changing the phase angle (and vice versa) is practically impossible, though. A change in phase angle always causes an instantaneous change in frequency and a change in frequency has to cause an instantaneous change of the phase angle. The word ‘instantaneous’ is very important as ‘rate of change’ and ‘instantaneous rate of change’ are two completely different things.
Frequency modulation and phase modulation have more in common than they are different. Both modulation techniques belong to the group of angle modulation techniques. Angle modulation – as the name implies – always affects the phase angle (and thus the frequency) of the signal.
With phase modulation, the change of the phase angle is proportional to the message that is to be modulated onto the signal. With frequency modulation, on the other hand, the instantaneous change in frequency is proportional to the message that is to be modulated onto the signal.
SPM(t) = Scarrier * cos(ωcarrier * t + Δφcarrier * cos(ωmessaget * φmessage))
SFM(t) = Scarrier * cos(ωcarrier * t + φcarrier + μ * cos(ωmessaget * φmessage))
Thus any FM-Modulator can be used to generate phase modulated signals if the message is being differentiated prior to modulation. The other way around a PM-Modulator can be used to generate a frequency modulated signal if the message is being integrated prior to modulation.