Gum drops restore some of your HP and cause no radiation poisoning, but they are very rare. While the box says "Radioactive", the gumdrops have a lower radiation count than other food items (1, as opposed to 3). Because of this, even a slight radiation resistance can negate this, rendering it in-game as 0. Gum drops are one of the rarest consumables in the game.
Gum drops are much more common in Fallout: New Vegas as opposed to their Fallout 3 counterparts, which were one of the rarest items in the game. Gum drops are used in the creation of Mushroom Cloud.
If the Courier has either gum drops or bubblegum in their inventory when talking to some of the Mini Boomers, an option appears to offer radioactive candy. If chosen, this will earn you Boomersinfamy and the player will no longer be able to pass a Charisma check with that Mini Boomer for fame. If the Charisma check has already been passed, this will not work, as that Mini Boomer will no longer have dialogue with you.