According to the full list of changes we received from a reliable source, the looping in the original music has been improved, and the instrumentation in the remixed music is now mixed properly (so you can hear bass and drums, for example). In addition, the "typing" text noise is no longer far too loud. In non-sound-related changes, GameCube controller support has been added, and the "extra" modes are now unlockable instead of open from the beginning. Typos in the translation have also been edited, and transparency has been added to the graphics in several places, including when you open up the Home menu.
NeoGAFfer sfried notes that if you've saved Cave Story to an SD card, you'll have to delete the game ("channel") and redownload it. Otherwise, you can just grab the update from the Wii Shop.