Well, normalizing raises the amplitude TO a certain level under maximum (e.g. -3 dB),
The amplifying function increases the level BY a certain value.
So when you normalize, the loudest part of your recording will determine how much the level is raised.
This means that if there is a loud crackling somewhere in the recording, an otherwise very quiet recording will only be amplified slightly or not at all.
That is why you should decide individually whether normalizing or amplifying leads to better results.
Regardless of this, it is important to adjust gain as precisely as possible during the recording, because in the later recording only 16 or 24 bits are available (If you normalize a very quiet recording, only few bits to represent the amplitude changes will be spread to 24 bit > valuable details are lost due to amplitude clipping).
Best way to get the perfect gain is to choose a recorder supporting 32 bit floating point.
Even with the weakest signals or the strongest level amplification, there is never any risk of the dynamics of the processed recording being impaired.