Without going into too much detail.
Tomahawks are not Amreican or Native American.
Native Americans --prior to Western or European contact--only had War Clubs, Stone Axes and Copper Axes.
Native Americans did not Smelt Iron so therefore did not have steel.
The first Tomahawks were most likely brought here by the French.
Axes , Hatchets and War Axes were common to the French culture prior to the 1st Century A.D. to as late as W.W. II. French Army Engineer units still issue an axe that looks like what we Call a Tomahawk.
The earliest Tomohawks were traded , By the French, to the Native American Tribes for Furs and Provisions.
Once the French established themselves here on the continent--The English also opened trade and exploration and brought Sheffield Made Hawks (Small Axes) with them also.
They were more than weapons and were used for light chopping and camp chores.
The Tomahaks became a trade item--a tool for mountian Men and a weapon for both Civillians--Native Americans and Some Early Military Units here on the Contenent.
Lewis & Clark ( 1805) had Hawks with them as tools--weapons and trade items.
The usefulness of a Hawk is directly related to the quality of the Hawk or the quality of the Hatchet you are comparing it too. Both can be useful tools and both can be effective weapons.
The hatchet will always Split wood better than a Hawk becaause there is a slower transition from the edge to the eye. The abrupt change from the edge--thru the face and to the eye of a hawk will stop the hawk in a log--rather than allow the head to smoothly open the split and continue down thru the log.
The Hawk will chop the same --better--or worse depending on the weight and geometry of both tools that you are comparing.
You have asked a Very--Very--Very technical Question that there is NO easy answer.
You can only start by giving the absolute specific two tools to be compared.
There is no general answer to your question.
Sorry I couldn't be more help but what I have said above.
There is a lot going on when you swing an axe at something and it's very difficult to address it in general terms.