I'll try and regurgitate some of the stuff from my camp craft course: The carpenters axe is for getting a nice even flat surface and can even be used almost like a plane, hence the completely straight blade. The broad axe has quite a heavy head and is mostly used for cutting across the grain, for instance lets say youve just split a 6 ft long log thats 1 foot across, you would lay the log down with the flat side pointing to the right and you would sit on the opposite side of the log, parallel with it and cut down across the grain, allowing you to look down across the log and see where the high points are. Like in picture 18 (if the link works)
http://www.raymears.com/Bushcraft_Courses/Photo_Gallery.cfm/id/43 only he is sitting on the same side that he is cutting you could just as easily sit on the other side of the log and look directly down and be able to see the surface more easily.
A carving axe usually has an upswept toe which allows you to take off chips a bit more accurately and also take off smaller chips, carving shapes that you may usually use a knife for. Also a feature on the carpenters and carving axe is for the cheek or ramp of the axe to be not as broad/a bit narrower, allowing you to grip the axe further up the haft, around the throat so that your hand is almost directly above the cutting edge, giving you more accuracy with each cut. Another defining feature for these axes is the helve shape; the swedish caving axe has a very curved shape which gives more of a swiping motion when you cut which slices the wood rather than just cutting straight down, whereas on the carpenters axe, it has a completely straight helve which encourages vertical cuts straight down, in more of a planing motion. As for the broad axe I think its somewhere in between but not sure
you can see these feature on the Gransfors Bruks swedish carving axe, GB carpenters axe and the GB swedish broad axe, all of which I have used on my camp craft course and fit their purpose very well.
As for hewing axes I havent a clue, I assumed hewing was just carving but I'm open to new knowledge on this.
Well there you go, happy to be corrected by anyone on this, as I'm sure there are many on here that know better than I do on this sort of thing but hope this helps a bit
paddy