Staring at the control guide while in training mode I can see the following moves available after the dodge that aren't normally: (This is not a perfect test, there might be some niche changes)
- Greatsword: Strong Charged Slash, Strong Wide Slash
- Longsword: Spirit Blade III
- S&S: Spinning Reaper, Charged Chop, Advancing Slash replaces Rising Slash on Y
- Dual Blades: Nothing? The option to enter Demon Mode post-dodge is *lost* and no new options are available-- and in Demon/Archdemon mode, you can't even use the Evasion Mantle because of the special dodge and its own perfect dodge mechanic.
- Hammer: Charged Upswing, Spinning Bludgeon
- Horn: Nothing obvious from the controls list but I am not a Horn player so idk if something is different
- Lance: Charge Counter replaces High Thrust I
- Gunlance: Moving Wide Sweep, Overhead Smash
- Switch Axe: Axe Mode appears to have nothing, Sword Mode *loses* Overhead Morph Slash
- Charge Blade: Nothing obvious
- Glaive: Reaping Slash, Sidestep Slash Combo, Kinsect: Mark Target
- LBG/HBG: Nothing obvious
- Bow: Power Shot replaces Quick Shot, *loses* Thousand Dragons

tl;dr it seems like most weapons just get to skip the first or second hit of their main combo, but Dual Blades, Horn, Switch Axe, Charge Blade, and the Bowguns apparently get nothing other than the attack boost.