What is the advantage of MATLAB's filtfilt
You can best look at it in the frequency domain. If $x[n]$ is the input sequence and $h[n]$ is the filter's impulse response, then the result of the first filter pass is

$$X(e^{j\omega})H(e^{j\omega})$$

with $X(e^{j\omega})$ and $H(e^{j\omega})$ the Fourier transforms of $x[n]$ and $h[n]$, respectively. Time reversal corresponds to replacing $\omega$ by $-\omega$ in the frequency domain, so after time-reversal we get

$$X(e^{-j\omega})H(e^{-j\omega})$$

The second filter pass corresponds to another multiplication with $H(e^{j\omega})$:

$$X(e^{-j\omega})H(e^{j\omega})H(e^{-j\omega})$$

which after time-reversal finally gives for the spectrum of the output signal

$$Y(e^{j\omega})=X(e^{j\omega})H(e^{j\omega})H(e^{-j\omega})=
X(e^{j\omega})|H(e^{j\omega})|^2\tag{1}$$

because for real-valued filter coefficients we have $H(e^{-j\omega})=H^{*}(e^{j\omega})$. Equation (1) shows that the output spectrum is obtained by filtering with a filter with frequency response $|H(e^{j\omega})|^2$, which is purely real-valued, i.e. its phase is zero and consequently there are no phase distortions.

This is the theory. In real-time processing there is of course quite a large delay because time-reversal only works if you allow a latency corresponding to the length of the input block. But this does not change the fact that there are no phase distortions, it's just an additional delay of the output data. For FIR filtering, this approach is not especially useful because you might as well define a new filter $\hat{h}[n]=h[n]*h[-n]$ and get the same result with ordinary filtering. It is more interesting to use this method with IIR filters, because they cannot have zero-phase (or linear phase, i.e. a pure delay).

In sum:


if you have or need an IIR filter and you want zero phase distortion, AND processing delay is no problem then this method is useful
if processing delay is an issue you shouldn't use it
if you have an FIR filter, you can easily compute a new FIR filter response which is equivalent to using this method. Note that with FIR filters an exactly linear phase can always be realized.