White LEDs are blue LEDs with a phosphor that absorbs some blue light and creates the other colours. That bit is not far different from fluorescent tubes that create UV light, but in fluorescent tubes all the visible light from them comes from the phosphor.
Multicolour or colour changing LEDs are a combination of blue, green and red LEDs that can each be driven separately to create the colour required.
I don't know if the CRI for a colour changing LED set to white would be better than the CRI for a white LED, but the spectrum would be different, so it could be worth looking at both.