I think the difference is this: a walking stick is an aid to walking, and replaces the much older walking staff. The cane is not an aid for walking, it replaces the sword and also the officer's baton and is one of the fashion marks of a 'gentleman' it's a class indicator rather than a practical item.
Interestingly, when swords started to fall out of fashion, many canes had a sword set inside that cuold be drawn or flicked out of the cane for use in a sticky situation.
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