It's not the same thing. Normalization is scanning to see what the highest volume scale you can use without introducing clipping, then multiplying every sample by that scale.

Compression is very different. The volume scale isn't constant, and the original sound is distorted significantly. I often use a compressor to listen to video game streams because they tend to have the game audio be way too quiet. Having the compressor on causes the game audio to become louder with some minor distortion, but distorts the streamer's voice significantly.

I know about this. This is why I asked