A non-fairy tale "Wish" is just a human desire or yearning and is open to anyone regardless of religion and the fulfillment of which doesn't necessarily carry a sense of urgency or obligation and doesn't necessarily assign the task to anyone or anything in particular.
A fairy-tale "Wish" are those you ask from a genie or from a wishing well. Everyone is free to disbelieve and everyone treats this activity in good fun.
A "Prayer" is the "serious" method taught by hucksters of an invisible/unscrutinizable product to request for things from this invisible/unscrutinizable product. It's almost like a fairy-tale "Wish", except that in this form of business, the customer is always wrong when the "prayer" fails.