When walking your foot is in contact with the ground up to 50% more than when running, for this and other reasons there are a number of subtle differences between the two shoe types. A walking shoe will often have a bevelled or angled heel due to the angle of foot strike when walking. Another key feature of a walking shoe is that it is designed to be especially flexible through the forefoot to enhance propulsion during toe-off.
Running shoes will generally be lighter in weight than a walking shoe and because they're designed for a higher impact will have more cushioning. For this reason it is not uncommon for people to use a running shoe for walking or a combination of the two. It is not recommended to use a walking shoe for running in for the reason mentioned above and because pronation is usually more excessive when running a walking shoe often is not supportive enough.
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