You will get two of the same item.
For example, if a plot activity could potentially grant you items A, B, or C, but you only earn one prize per day, that means that without the premium perk (or a fortune cookie), you will only get item A, or item B, or item C.
Say you get Item A for completing the plot activity. With the Double Plot Prizes perk or Fortune Cookie, you will be granted two (2) of item A!