Limiting gets rid of the AM component of the noise but not the FM
component, so by itself it improves the SNR by only 3 dB in a narrowband
system.
Besides suppressing noise, capture prevents distant stations or adjacent
channels from changing the instantaneous average frequency of the
carrier. When you add two phasors of different lengths and similar
frequencies, the average frequency with which it loops round the origin
is just that of the larger one. The smaller one can make it bounce back
and forth, but can't make it loop faster or slower.
Normally, the detected noise density goes up as you increase the IF
bandwidth, because all the noise components intermodulate with each
other in the detector, and some of that creates baseband products.
Interestingly, because of the large carrier amplitude, the noise doesn't
intermodulate with itself much, but instead is linearly downconverted at
the frequency discriminator's output. What this does in the frequency
domain is to confine the relevant noise bandwidth to +-15 kHz of the
instantaneous carrier frequency. Thus given a fixed baseband lowpass
filter, the detected noise doesn't go up as you widen the IF, and the
detected signal gets bigger as you increase the deviation, all assuming
that you stay in the high-SNR limit. That's a cool thing that I just
learned today, from Armstrong's paper (thanks, George!). (I had thought
that the noise voltage went up like sqrt(BW), but it doesn't.)
Floyd Gardner's PLL book has a good section on loops with limiters. The
math gets fairly hairy.