The main tone difference is the the neck woods. The Brown Bass has Mahogany as the primary wood, SC Deluxe has Maple. The other woods contribute too, but the neck sets the stage, so to speak.
So the Brown Bass has an overall more warm and round tone as its default, and it can make crazy bright tones. The SC will have a brighter default tone that you can pare down with lower filter settings, but it won't get as round sounding as the Brown Bass.
I've written a little more detailed explanation of the
neck woods you might want to explore if you haven't seen this relatively new page yet.
Perhaps a player that has both might chime in with their observations as well.