According to a Capcom-Unity post, the "Son of Sparda" mode changes enemy placements and increases enemy difficulty, including new enemy behaviors. The "Dante Must Die" mode starts with that, and adds "the strongest enemies and insane attack waves."
"Heaven or Hell" mixes it up further by enabling one-hit kills on everyone in the game, including Dante. And "Hell and Hell" is the same mode, with only Dante susceptible to one-hit death. The common thread in all these new experiences: you'll die over and over again.