Is it the amount of padding in the nose of the saddle?

Yup...and maybe some added strength to the front as riders tend to get out on the nose of the Tri/TT saddles more than a normal saddle.

And if you're going to be riding on aerobars in an aero position they are worth the slight weight penalty. A lot more comfortable, and it also gives you the option of riding very far foward, making for less extreme angle at your hips. I find it nice to be able to sit on the nose for a while, just to have a break, and still be aero.

I like to sit further forward on the saddle when climbing. Would this be a good option? How do they feel when cruising along on the hoods?

..and if you are doing a triathlon, you are not riding in bike shorts. Not unless you want to carry about 15 pounds of water in your bike chamois...

the extra padding helps with the lack of what tri shorts provide.

I don't get it. Unless you doing the ironman, the bike portion isn't that long. Should comfort really matter that much in a race of less than an hour. In training, unless you are doing blocks, you can still wear bike shorts.

My chamois are barely padded at all and my saddle, a brooks, has zero padding.

tri shorts, and tri suits are designed with a relatively thin chamois thats made to avoid retaining water, but they still have some padding down there.

Can't speak to generalities, but my tri saddle fits like a torture device with a full chamois. With my tri shorts, it is very comfortable, especially on the nose. I think long story short, there is probably a cumulative amount of padding that can be added together for the saddle and chamois which results in comfort for most people. Is this important in tri - yes....with my cycing shorts, it is excruciating to ride even a mile or 2 in the aero position....I cant apply any power to the pedals and my perrinium feels like it is going to burst. So yes, comfort is important. What you have to remember is that in a forward aero position, you are in a much more aggressive position than on a road bike, even in racing geometries, and it required a much more athletic position relative to the road.

Errrr. . . . a tri saddle might be 50 grams heavier than an average road saddle due to the extra "padding." I don't think that's worth giving up on comfort. Get on a tri/tt bike with really low handlebars sometime - it's nowhere near the same position as on a road bike, padding (or at least a comfort-oriented shape) is more critical. Your entire body is rotated forwards, so you really have no chance of putting weight on your "sit bones" as you would when riding a road bike. Like it or not, your body weight is pretty much all going to be concentrated directly on your genitals. Getting into this position on a regular road saddle or the wrong tri saddle, even with padded shorts, can be totally, unnervingly, uncomfortable in 5 minutes, let alone even a 10 or 15 mile sprint triathlon distance. Even if you were "tough" enough to make it through the bike portion with an uncomfortable saddle, I don't think I'd want to continue on to the run portion with a sore crotch.

To the OP - besides padding on the nose, the overall shape of tri saddles is typically very different. They're usually much wider up front (to help distribute weight when you're out on the nose) and usually less wide out back in the wings (since you're almost never on the back of the saddle anyways.) To sum up, a "regular" road bike saddle is generally designed for you to put pressure on the wider wings near the back of the saddle, with your sit bones. A tri saddle is designed to make the best of a position that's naturally much less comfortable - putting weight on the nose of the saddle through a spot farther forward on your anatomy.

That said, they can be comfy if set up correctly. I have a Selle San Marco Azoto Tri Race Gel saddle, and my position lets me get aero while staying far enough back on the saddle to be pretty comfortable. (This saddle is 280g, FWIW.) If you're a true weight weenie, Selle also makes a 190g saddle, the Selle San Marco Aspide Black Angel. I've never ridden one, but my 290g saddle hasn't ever prevented me from doing well and having fun.

I was going to answer, but it appears that's been covered.