The fights are more difficult. Things tend to hit harder/have more health than the normal/story mode counterparts.

Some times there may be different/added mechanics to the Savage fights compared to normal fights. For example: A11N has telegraphed AOEs/attacks where as A11S does not have telegraphed AOE markers. Both versions of Sophia have the tilts, but the tilts are far more forgiving in the Hard Mode version.

Savage fights usually have an Enrage mechanic that happens sooner than a normal mode counterpart (if the normal mode has an Enrage).

I'm sure there's more differences, but these are ones I can think of off the top of my head. Generally, Savage doesn't have a storyline of its own.