From my experience in the American metal fabricating industry, I would say that the two terms are often used interchangeably, but not always.
A wooden platform for stacking and moving product with a forklift or pallet jack could be called a pallet or a skid. They are equivalent when speaking generally without reference to a specific sub-type: "How many pallets/skids will fit on that truck?"
A steel item of similar shape and purpose is almost always a skid or a metal skid. Those would typically be for in-plant use because of their expense and great durability.
A plastic item of this class would usually be called a pallet or recyclable pallet. Recyclable pallet can actually mean two different things: a plastic or wood platform that is supposed to be returned to a supplier to be used again, or a wooden pallet that was salvaged, repaired, and resold at a discounted rate.
The tool pallet jack is common, but skid jack is almost unheard of.
Pallet can be be made into a verb as palletize, but skid rarely becomes skidded.