The question 'are FM and PM the same or different?' has two answers ...
They are the same
They are both angle modulations of the carrier. If you see an angle-modulated carrier, then you have no way of knowing, without other information, whether you are looking at an 'FM' or 'PM' system.
In principle, any carrier produced by one system can also be produced by the other, subject to the practical limitations below.
They are different
They were invented at different times to serve different purposes, so a 'PM' system will carry with it a different set of assumptions to an 'FM' system, which will usually make a typical signal from either look different.
There are at least two ways to get angle modulations of the carrier. We can generate a fixed carrier and then phase modulate it. Or we can generate a carrier with the correct average frequency, while frequency modulating it.
These two methods tend to produce systems with a different mix of characteristics.
Generally a phase modulator can only be economically built to produce modulations up to a radian or so. With the modulation thus limited, the resulting carrier would fit well into a narrow-band low quality analogue communication network. It's not possible to produce many cycles of FM modulation, at least not directly. As angle is mulitplied when the frequency is multiplied, you could use a +/- 1 radian phase modulator to produce +/- 10 radians by passing the multiplied signal through a frequency multiplier.
A frequency modulator can easily produce angle modulations of many radians, many cycles even, by using a voltage controlled oscillator (VCO). This improves the SNR, and wide deviation FM is used for high quality analogue music transmission, albeit with very wide channels. Uncertain control of DC drift to the VCO means that it would not be possible to return to exactly the same phase of carrier, which a PM system could do, without extra control by a synthesiser.
A significant difference is the 1/s difference in modulation response, as phase is the integral of frequency. This can be mimicked by pre- or de-emphasis on the modulation signal into or out of the modulator or demodulator respectively, so it's not a fundamental difference between the systems.
You can construct a universal angle modulator, where a fixed carrier is IQ modulated with a complex baseband signal. With suitable generation of I and Q, usually in an FPGA or ASIC, the output can have definite phase, many cycles of modulation, and produce signals identical to any FM or PM system. However, this is never done for real FM or PM systems; once you have this level of control, you might as well use the amplitude component as well, to carry more information, and control the channel width better.
