Most folks aren't extraordinarily dirty, so bath water won't get too nasty. Actually, the opposite issue occurs when the soap remains in the water/on your skin when you get up to dry off.
A shower can use less water if you don't shave or brush teeth in it. You can buy a cut off valve for $5 at a hardware store to cut flow of water in shower if you do to save water.
Shower can blast continual hot water on the skin which can strip too much oils if you get dry skin in winter. Soaking in a bath doesn't strip your skin oils as much.
Bath can be slow, but kids that don't know how to shower can stand in there wasting all the hot water. In a tub, you can at least regulate how much water is dispensed for your kids to use.
The Japanese are big proponents of baths and water rinsing. It keeps the skin from getting to stripped from continual hot water.
There's pro cons to each, so you have to decide which angle you're concerned with (health, water conservation, etc).