Each tool has attributes that will tell you what it's good for:
- Saws have precision and clean, straight cuts.
- Axes have mass and momentum.
- Hatchets have the same intentions as an axe, but with less mass and a shorter handle giving them a bit more control.
Felling is the act of cutting down a standing tree. Lumberjacks use huge, powerful chainsaws for this, and if you have one of those, you probably know more than me about this topic. For the rest of us, cutting down a tree with a hand saw can be an endeavor: as you cut through, the tree can lean into the cut and bind your blade. On limbs and small trees, this isn't a big problem; just push the tree or branch away to keep the blade free. On standing material larger than about 4", though, this becomes more difficult.
An ax cuts with a different action, and won't suffer from this binding. It requires more practice and more physical effort, but is ultimately safer and easier. A hatchet can work for very small trees, but you`'ll work yourself to death on anything of size.
Once the wood is on the ground, saws show their strength because your material is no longer working against you by trying to fall to the ground. With a saw, you can break your wood down to lengths that work for a fire, or to a needed length for building material. A saw also gives clean, square ends, where an ax does not, which can be useful for building things in a camp. An ax can perform these tasks, but it takes more time and energy.
Many bushcraft constructs require notching your building material. A skilled axman makes this look effortless, and with skill and practice it can be, but without that skill and practice you'll drive yourself to frustration. The rest of us should use a saw, which cuts precise notches with very little practice. It is a slower process, but it works. Practice with the ax or hatchet, but know the saw will fill the gap until then.
When you're trying to make round logs into not-round logs, the mass and leverage of an ax makes it the only way to fly. While a saw could theortically do the job, that would require so much more time that doing so is beyond impractical. If you want to work with smaller wood, such as kindling, the greater control of a hatchet or tomahawk makes it ideal for the task.
There is no single do-all tool -- but now you can pick the right tool for the task.