This is an infestation of Crape Myrtle Bark Scale. One of the first signs of this insect is black sooty mold on the bark. Bark and leaves may feel sticky to the touch as a byproduct of where the scale is feeding. The "white spots" can occur anywhere but usually in the branch crotch or pruning wounds.
Treatments for this scale include using a mild solution of dishwashing detergent and a brush to scrub the scale and black mold off. Horticulture oil has not been shown to be effective yet but a heavy winter coating might help. Full coverage is a necessity. Systemic insecticide treatments in the form of a soil drench have shown the most promise to date. These include Imidacloprid and dinotefuran. The best time to treat with the systemic insecticides is May through July. Follow the directions for use for either product. Systemics do take a few weeks to become effective as they spread throughout the plant