Well, I've owned and built 3 different TJ's (one Rubicon 2 door) before I bought an H3 adv/lux. The H3 is a nicer rig no doubt. It doesn't rattle, the interior is way beyond what a wrangler offers. I've been following the JK though.
Trail rides aside, If rockcrawling ( a fast growing sport) is your thing, you'd best go the JK. The JK (rubi ed.) was flattly designed for this in mind. The jeep engineers listened to the jeep owner's thoughts on how to make the TJ better and they really did a great job on the JK. You'll never see an H3 make it through the Hammer's in JV. You'll see tons of JKs there in no time flat. With a simple swap of coils (4 lift) and shocks basically on the JK you can stuff in 37" tires! Jeeps are easier to re-gear (and find gears) for those tires too! There's other things, nothing really hangs below the frame rails now on JKs, the gas tank has been relocated, frame is even stronger now. Longer control arms will help it climb and articulate much better than the TJ did. Front locker too.
However, if you go JK, you'll never get the same ride of an IFS front rig, and you'll never see a wrangler with leather from the factory!
The H3 is a very capable 4x4, but just isn't designed with rockcrawling (Hammer's style) in mind, even with a lift. Its wider, longer and the front diff (lacking a locker too) just isn't strong enough in my opinion for that kind of abuse. I can't imagine lifting a H3 onto 37" tires, and trying to do heavy rockcrawling, that front diff will pop fairly easy, and easier if you regear (pinion will get weak with a regear due to size). Throw in a front locker and those 37" tires, i'd bet it busts on the first foot tall rock you try to climb. It wasn't designed for that. They may solve this with the H3 alpha from what others are saying though.
I'll never part with my H3 (except for an alpha!). Its just too nice of a ride, no rattles, nice interior and still plenty capable to get me through nasty snowy/icy roads. It beats the crap out of any landrover, yota, BMW, Porche, lexus, Merc, and even jeep grand! But in the very serious rockcrawling world, you can't beat a Wrangler, you just don't get some of the creature features like leather and the buff hummer look. I also can't stand the new fenders on the JK. Lost on the retro look I guess.