You’ve been saying it in the wrong context forever and now it’s time to stop.
What you think they mean: The same thing.
What they actually mean: The palate is the roof of the mouth and also a person’s ability to discern different flavours, while a palette is what an artist uses to mix paints.
Neither are to be confused with pallet, which is a wooden platform used to stack things.
What you think it means: To deny something.
What it actually means: To prove a statement or theory to be untrue.