If you use peak normalization rather than RMS normalization, all you are doing is just bringing up everything equally. Just like turning up the volume. If you use RMS normalization, you're actually changing the dynamics of the original source. Here is an explanation from a list I'm on, though he's talking about orchestral music, the ideas still apply.

I'm back. I just thought I should have probably better explained the
difference between peak and RMS levels. As I said, the peak is the highest
single level in a file. But due to the potentially transient nature of
peaks, they can be quite misleading. For instance, You may have a piece
wherein the highest relative peak is actually something like -4 dB. But, and
here's where it gets tricky. At the end of your symphonic piece you may have
a crescendo which ends with a burst of sound from the entire orchestra. So
that and only that one single peak time may hit 0 or close to it. Analyzing
this can reveal what you have to do; compress or limit that peak to get a
more realistic read on the file. So let's say you compress just that last
measure down to a -2 dB peak. Now you've got 2 dB more headroom to work
with. This in turn gives you 2 dB more of the perceived volume (RMS) you can
raise before the compressor kicks in to protect against overloads. And as
long as it's not too severe, you won't get the perception of squashed
dynamics.

In other words, you can look at the stat's of two files, both having a peak
of 0 dB. But one having an RMS of -21 and the other having an RMS of -18 dB.
So if done right, you can have your cake and eat it too.