I'm assuming it's the same as in the UK from your description? So Hack or Lightweight Hunter Under Saddle classes?
Hacks are generally smaller, lighter, more thoroughbred-y types with very showy movement. Manners are vital in a hack class, on top of the movement, but the ideal hack will be smaller than 16hh and very light. Used to be shown sidesaddle, they are considered a lady's ride.
Typical hack type:
Hunters, even lightweight hunters, have more bone, and tend not to be TB - they are designed to carry up to 12 stone in a lightweight class, and whilst you wouldn't jump them in a flat ridden class, they should be built to carry you for a days hunting and jumping.
I love hunter shows, and took my old horse, who was a middleweight hunter (ID x Welsh) to a lot of hunter shows. My latest horse, though, is a 16.2hh TB, who doesn't really fit into any of these classes - he's too tall and not quite fine enough for a hack (to place well, anyway) and far too light and not enough bone for a hunter. However, he gives the judge a very good ride, so he occasionally places based on that.
Not going to lie, here, a 16.2hh TB with any amount of bone is unlikely to do well in either class unless they are in pristine condition and ride like a dream with stunning paces. I don't know your boy that well, and obviously with weight and training he could potentially get there, but I personally wouldn't set your heart on those types of classes... Trust me, I know how disappointing it is! However, if he can jump, you could always look at working hunter classes. That's what we do instead - a greater percentage of the marks is on the jumping portion, and we pick up the points there that he loses for not having the hunter build.
However, always worth entering either or both at a few local shows and see what the competition is like - could be different where you are, and possibly not as strict at local level. Plus great fun and a low-stress way to introduce your horse to buzzy competition environments without asking more than w/t/c/g...
And just out of interest, a heavyweight hunter, because they are awesome:
This shows what a judge is looking for in a less exaggerated scale in the lighter weight classes... this sort of build.