I've watched a few videos on cutting tools and the use of axes. However I'm not clear on what the advantage (or disadvantage) is of a tomahawk style axe over a typical belt or hand axe, such as this estwing.
Unfortunately the words "Tomahawk", "Hand Axe", "Hatchet", etc are not rigorously defined terms. If I wanted to make my own heavy blade attached to the end of a stick and sell it, I could use any of those terms.
That said, typically a Tomahawk has a lighter head on a longer handle (and usually a smaller cutting surface, because it's a smaller head). A Hand Axe has a larger, and heavier, head on a shorter handle. A plain-jane axe has a heavy head on a much longer handle.
As for trade-offs, it comes down to the weight you can carry, the type of cutting you need to do, and the accuracy with which you need to accomplish your task. I find any axe/hatchet/tomahawk with a lighter head and longer handle makes accurate cuts easier to make, while the heavier head on the shorter body allows for more powerful cuts, with the most powerful cut coming from a heavy head on a long handle (aka your typical axe).
If you are showing off your axe throwing ability, a tomahawk (as defined in this answer) is better since it will spin less. Otherwise, if you are trying to actually chop wood for useful purposes, the hand axe will make quicker work for you.