General problems and benefits (which apply equally to adults and children):
- Chance to slip and fall (causing injury)
- Uneven distribution of water over body / having to step out of the water (not really a health risk, but can cause discomfort); this will depend on type of shower head, where the head points, height of the person, and size of the person.
- Usually much quicker.
- Little/no prep time required
- Dirt and such is rinsed off and directly down the drain.
- Chance of drowning (falling asleep, passing out, slip/fall)
- Dirt, soap suds, and such stay in the bath as you wash (bathing in dirty water)
- Long prep time (which can be more difficult to get the 'right' temperature)
- Water cools over time (this may be a detriment or benefit depending on opinion)
- More relaxing (dependent on personality of bather)
- Possibly beneficial for sore muscles or the like (bad backs?)
For younger children, make sure they're being observed regardless of whether they take a shower or bath. Overall, the only reason really to definitely take a shower over a bath is dependent on how dirty the bather is. If the bather is very dirty, a shower is better due to the benefit of having all the dirt and such washed off and straight down the drain. Otherwise it's almost completely dependent on the personality of the bather and possibly any time constraints you have. (This is assuming that the bather is competent enough not to have an accident like slip/fall).
I looked to see if I could find research proving that either of the choices were healthier (common sense told me that a shower should be healthier due to not 'stewing in your own filth') but was unable to find any conclusive research one way or the other. Either the topic hasn't been well researched, or the research that has been done has found no significant health benefit/risk between the two. The only exception being the possible benefit of a hot/cold bath on sore muscles and such.