Two main things about the hatchet and the (small packing) saw.
A saw takes less effort to cut smaller branches, and wood and is safer to use (less likely to hurt yourself and others) and makes a MUCH cleaner cut. A small packing saw can be used to cut exacting shapes into wood for construction.
The hatchet (sometimes) can be faster, and will cut MUCH thicker pieces of wood and small trees, the hatchet also makes a VERY effective weapon.
In many of the more popular campgrounds the axe and hatchet are forbidden for use.
We were going to return to the Kukri, Golok, Parang, Machete and Bolo issue and I stumbled across this video that clearly shows the saw, hatchet, Kukri in action, and the time to chop was almost exactly the same from hatchet to Kukri.
The folding saw for safety, very good points.
The Fiskars hatchet
The Cold steel shovel as a hatchet - this is what we have switched to for packing (dual use).