On harder wood, a tomahawk will generally outchop a machete. On brush and other lighter/softer foliage, the machete would usually have the advantage. Also, it can be easier to choke up on the hawk for more detailed cutting duties.
You should also consider the khukuri, hatchet, large camp/bowie knife, and Uluchet (
www.silverstar.com/turnermfg). I've used a light khukuri on everything from brush to 2" limbs with single stroke cutting. On limbs over about 2", a couple chops might be required. I really like them.
For hatchets, the Gransfors Bruks Wildlife Hatchet is about the best I've used. Hand forged with great edge holding ability.
I've also used the Busse Basic 9 and large knives of similar style for brush abatement and chopping. Good, but not as effective as many khukuris for this application.
The Uluchet is also a handy tool if chopping is secondary to other cutting and skinning chores. It is razor sharp D2 with impressive edge retention and the ability to lengthen/shorten the grip as needed. Very useful for hunters.
Test the hawk under your conditions and see how it compares with your machete and any other available options. Please let us know what your results are.
Oops, I forgot: HAPPY BIRTHDAY!