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.
Compression isn't just gain adjustment - it's a specific type of audio processing that increases perceived "gain" (loudness) of the entire source audio by "compressing" the levels of loud frequencies & increasing the levels of quiet frequencies.
Normalization increases gain of all frequencies at any given point-in-time while reducing gain of all frequencies at other points in time. It doesn't reduce dynamic range.
