Palate is the roof of the mouth, and by extension your sense of taste — a refined palate. Palette is the board an artist mixes colours on, and by extension any range of colours. A third spelling, pallet, means a wooden shipping platform or a crude bed.
The core difference
Three homophones, three unrelated meanings.
- palate — roof of the mouth; sense of taste: a delicate palate.
- palette — artist's mixing board; a range of colours: a muted palette.
- pallet — a shipping platform, or a straw mattress.
A way to keep them straight
Palette looks French and is — the extra -tte goes with art. Palate is the anatomical one and shares its ending with other body-part words from Latin. Pallet, the plainest spelling, is the plainest object.
Figurative reach
Both of the first two extend widely: a palate for wine, a colour palette, and by extension a palette of options or techniques. Those figurative uses are standard, which is part of why the spellings get mixed.