What is the difference between frequency modulation and phase modulation?
A signal that is phase-shift modulated with some continuous signal f will have the same waveform as one which is frequency-shift modulated with the derivative of f. Likewise, a signal that is frequency-modulated by some signal f will have the same waveform as one which is phase-modulated by the integral of f. Note that in some cases, it may not be practical to take the integral or derivative of f, so some signals can only be meaningfully represent phase modulation and others may only meaningfully represent frequency modulation.
Even though the waveforms of a frequency-modulated signal and phase-shift-modulated signal may look similar, however, there's often another important distinction. Phase-shift modulated signals are often modulated relative to some other reference signal. In some cases, the reference may be a signal sent on another wire. In other cases, the reference may be a signal which was sent at an earlier time on the same wire. In NTSC video, for example, every scan line starts with a few cycles of a 3.579545Mhz reference sine wave. Colors later in the line will be encoded by waves that have a certain phase relationship to that reference wave. Yellow, for example, will be represented by a wave whose phase matches the reference; blue will be represented by a wave 180 degrees out of phase. Red and green are represented by waves +/- 90 degrees out of phase. Note that all solid colors are be represented by the same frequency; the only difference between them is the phase relative to the reference wave.
Modulation in radio communication means, changing a parameter of the carrier wave according to the amplitude of the modulating signal. Generally modulating signal (often audio or video) signal is of a lower frequency and known as baseband signal. While a carrier signal (much higher in frequency, able to be transmitted with feasible antenna) is known as passband signal.
After this background, lets jump directly into the question. Frequency and phase are two different parameters of carrier signal. if we represent the carrier signal as c(t) = A sin (w(t)t + p(t)), w(t) is the (angular) frequency and p(t) is the phase.
Modulation means, we can vary either w or p or both in time according to the amplitude value of the modulating signal. Lets represent modulating signal as m(t).
c(t) = A sin (w(t)t + p)
w(t) = some_function(m(t))
c(t) = A sin (wt + p(t))
p(t) = some_function(m(t))
So, although the modulated signals resulting from FM and PM look very similar, they are fundamentally different.
Frequency modulation is using different frequencies to represent different pieces of data. The binary cases is probably the easiest to understand, in this case Frequency X would correspond to a binary 1 while Frequency Y might correspond to a binary 0. So from a receivers perspective, if you see power at frequency y then you put a 0 down, while if you see power at frequency x then you put a 1 down. More complicated FM modulators can use different frequencies and combinations of frequencies to mean different symbols. For example, you could have it set up such that if you see frequency x but not y then your symbol is 10, if you see y but not x then it is 01, if you see both then it is 11, if you see neither then it is 00.
Phase modulation on the other hand uses a fixed frequency. It is then the change of phase that tells the demodulator what data to output. It usually makes most sense to look at what is called a constellation:
If you think of this as each point on the circle being a different phase change, you can see how each dot could represent different data.