White's illusion is shown with two rectangles, named accordingly A and B. Both rectangles are covered with black horizontal stripes. Rectangle A is shown in between the black stripes while rectangle B is shown on top of the black stripes. This creates the illusion that rectangle A is darker than rectangle B. However, both rectangles are printed from the same color ink and when the black horizontal stripes are removed, the observer is able to notice they are actually the same color.
Stimultaneous contrast occurs when our perception of the brightness or color of one area is affected by the presence of an adjancent or surrounding area. This is most intense when when the two colors are complementary colors such as in White's illusion, black and gray.
Below is a video posted to demonstrate simultaneous contrast. Notice how the square in the center remains the same color throughout the clip: