As far as horticulture is concerned, there is no exact definition of when a plant is a shrub or a bush. A good general description of a shrub is a "woody plant with several perennial stems that may be erect or may lay close to the ground. It will usually have a height less than 13 feet and stems no more than about three inches in diameter." (from "What is a Tree?")
Many people use both "shrubs" and "bushes" to describe their plants. What's the difference, then? Overall, it's really just a matter of personal preference and regional language.
For instance, when I think of shrubs, I consider those to be specimens that are cultivated in a garden, and use that as the proper horticultural term. Bushes, on the other hand, are those plants out in the wild that fit the definition of a shrub.
Even my own definition is not universal, though. For example, I have worked with both rose bushes and butterfly bushes. I have never called them rose shrubs or butterfly shrubs, even though according to my own definition, they would be shrubs.
Others have different ideas of the difference between shrubs and bushes. The Oxford Dictionary says that a bush can be a shrub, or also a whole cluster (thicket) of shrubs or shrub-like trees. Others may say that a shrub is smaller and out in the wild, or similar variations.
Confused? Don't be. In the end, all you need to know is that there is no widely accepted difference between shrubs and bushes.