When you say taking a bath, this means that you would need to soak your body in a tub of water so that you can clean yourself. You would usually need to have some items so that you can clean yourself properly.
When you say that you are taking a shower, this means that you are going to clean yourself with the use of a showerhead that is usually placed a few inches above your head. It will spray you with water as you do the usual things so that you will be clean. You can shampoo your hair, use your different body products in order to be cleaner. Showers are more recommended when you are dirty or sweaty.