Easy Mode is a difficulty setting available in all Nicalis ports of Cave Story. In Easy Mode, all damage taken is halved. Quote will have a yellow outfit on, instead of his normal red outfit.
If the original damage that an enemy deals is an odd integer, it will be rounded down (e.g. 5 -> 2). If an enemy deals 1 damage originally, it will still deal 1 damage in Easy Mode.
In Curly's Story, her outfit will also change to yellow if Easy Mode is selected. Other than that, things are pretty much the same.
When you get the Arms Barrier in the Labyrinth in this mode, almost barely your weapon will level down by 2 hits.