What is the difference between carpet padding and underlay?

Nothing. They are the same cushioning layer installed between the subfloor and the carpet; "padding" is the more common term in Ontario, "underlay" turns up in some manufacturer literature and older trade documents. Whatever it is called, its job is the same, cushioning the pile, insulating the floor and absorbing wear, and the weight and density you choose still matters more than which word is on the invoice.