Although related, these three nouns have very different meanings—all of them refer to surfaces that you put something on.
Pallet refers to a simple bed such as a home-made mattress, or wooden platform:
She preferred to sleep on a simple pallet stuffed with straw. (bed)
Pallet can also describe a wooden platform used to carry things:
The men carried the pallets, loaded with goods, into the back of the store. (wooden platform)
Pallette can describe either the surface on which a painter holds paints (see below), or a range of colors, such as a painter uses.
Pallette can also be used to describe a range of flavors, sounds, or other sensations.
Palate refers to a part of the body—“the roof of the mouth”—or a person’s sense of taste:
The hot pizza burned his palate.
Kate had a sensitive palate for wine.
Also, palate can be used for any ‘sense of taste,’ such as taste in movies or music, as well as flavors.