Well, you can't just take colors per screen as the measurement. That only works for bitmap displays (like PCs and MACs), not tile based displays like the NeoGeo and other consoles. For tile based displays, it's the number of colors per tile + the number of palettes available.
For the NeoGeo it's a sprite based system only (with the exception of a single not scrolling window frame). Sprites are used to build the BG layers. Each 16x16/16x32 sprite can only access 16 colors with one color being unchange able(for transparency). The system has 256 palettes (and an altered bank of 256 palettes which you can switch, but not sure if this is mid screen or during vblank) - each palette entry is 32 reds, 32 greens, 32 blues for a total of 32k color master palette. So you have 3840 colors in 256 palettes from 32k of colors, with each 16x16/16x32 accessing 15 colors.
That's one reason for dithering, the other reason is that it adds "texture" to an image. SNES was guilty of pillow shading and looks kind of fake, dithering helps break that up. Dithering works great for colors/hues that are close together and the NeoGeo games look great with it IMO. Dithering on the Genesis stood out more and a lot of times looked terrible because the dithering between colors/hues were too great in difference.