When light or matter is consumed by the overwhelming gravitational pull of a black hole, neither has any chance of escaping. In a white hole the opposite is true; light and matter can escape, but there's no chance of them getting back in.
What's interesting is that you've actually simulated the latter scenario in your kitchen sink countless times, through a phenomenon known in the field of fluid dynamics as a "hydraulic jump."
Hydraulic jumps (various incarnations of which are pictured above) are pretty awesome in their own right; fuckyeahfluiddynamics explains:
Hydraulic jumps occur when a fast-moving fluid enters a region of slow-moving fluid and transfers its kinetic energy into potential energy by increasing its elevation. For a steady falling jet, this usually causes the formation of a circular hydraulic jump-that distinctive ring you see in the bottom of your kitchen sink. But circles aren't the only shape a hydraulic jump can take, particularly in more viscous fluids than water. In these fluids, surface tension instabilities can break the symmetry of the hydraulic jump, leading to an array of polygonal and clover-like shapes. (Photo credits: J. W. M. Bush et al.)
Hydraulic jumps have fascinated scientists for at least a century; and in 2010, researchers managed to demonstrate that these jumps are actually analogous to event horizons at the boundary of white holes.
When a stream of tap water hits the flat surface of the sink, it spreads out into a thin disc bounded by a raised lip, called the hydraulic jump… More recently, physicists have suggested that, if the water waves inside the disc move faster than the waves outside, the jump could serve as an analogue event horizon. Water can approach the ring from outside, but it can't get in.
"The jump would therefore constitute a one-directional membrane or white hole," wrote physicist Gil Jannes and Germain Rousseaux of the University of Nice Sophia Antipolis in France and colleagues in a study on ArXiv Oct. 8. "Surface waves outside the jump cannot penetrate in the inner region; they are trapped outside in precisely the same sense as light is trapped inside a black hole."
The analogy is not just surface-deep. The math describing both situations is exactly equivalent.
Read more about the researchers' experiment over at Wired.
3-sided clover (Bush et al. 2006)
4-sided polygon (Bush et al. 2006)
8-sided star (Bush et al. 2006)
Cat's eye (Bush et al. 2006)
5-sided polygon (Bush et al. 2006)