The effect of ``sticky'' additives (viz., those that have attractive interactions with the polymer) on polymer crystallization, has been investigated by dynamic Monte Carlo (DMC) simulations. Additive-polymer attractive interactions result in a slowing down of the polymer chain diffusivity in the melt state. Our results show that with increasing additive stickiness, polymer crystallinity decreases monotonically, and thinner crystallites form, viz., crystallization is inhibited by the presence of sticky additives. Unusually, the observed ``specific heat'' peak at the phase transition shows nonmonotonic behavior with additive stickiness, and exhibits a maximum for intermediate values of additive stickiness. While the origins of this unexpected behavior are not clear, we show that it correlates with a large interchange between crystalline and amorphous states of the monomers, in the vicinity of the additives. At this intermediate additive stickiness, we also find that crystallization follows a qualitatively different route-crystallinity shows a non-Avrami-like evolution, unlike the case at low or high additive stickiness.