When someone is sharpening the saw, they are taking time out, learning and improving so that they can be more effective in their job.
When someone has an axe to grind, they are angry, and looking for an excuse to take it out on someone.
Now perhaps the former image is that of the careful thoughtful craftsman. Perhaps the second image is one of sparks flying off a piece of metal in a grinder – but they seem too similar not to have some common origin.
My question is: Why do sharpening the saw and has an axe to grind mean two completely different things?