The map screen is no longer, each area is connected via savepoints that you can travel between once the second one is found.
Save files have been added.
Stage 1 has a mostly different layout and name along with stage 3.
You can perfom a strong attack while crouching, and the air version is different. The light attack no longer perfoms strong attacks when holding forward.

Paint can now be aimed with a crosshair when holding the strong attack key. You can't aim it in midair however.
MP now shows the values above the meter and anima crystals now increase MP intead of the recovery time.
There are key items that have one time uses and are shown on the pause screen.
Shop items are more expensive.
Power ups are harder to find on an initial playthrough of a stage.
Clothing resets only if you die or fail, if the options are set to no clothing damage restarting from a savepoint will reset it.
The treasure dungeon minigame is now hidden somewhere else.
The boss arena and fight has some changes to it.
There's more new stuff to find, but those are the most important things I wanted to mention.