So, tensioned leather saddles differ pretty massively in construction from modern saddles.

Modern saddles are a rigid piece of material (i.e. plastic) that has been upholstered. They are easy to manufacture into arbitrary shapes with arbitrary padding profiles, they're lightweight, and they can be pretty cheap.
Too little padding, and it might be too hard on your sit bones; too much padding, and your sit bones will sink too deep and fail to suspend softer tissues above the padding, resulting in said softer tissues being crushed.

Tensioned leather saddles work by suspending a piece of material between the front and back of the saddle. The surface is smooth and hard with no padding, but its overall structure has some amount of give and flex. This means that sit bones do not sink deep into the surface, but it also isn't overly harsh. The smoothness, flexing, and good breathability of these saddles can also reduce chafing.
The obvious drawbacks of tensioned leather saddles are that they're heavy and usually quite expensive (and there's a tiny bit of maintenance involved in caring for them, and there can be a breaking-in period). The less obvious, but very significant, drawback is that shaping is a tricky matter. Lots of people have issues with leather saddles having overly-prominent noses, for instance. A common symptom might be that a few postures are perfectly comfy, but a few aren't really usable. If you find a leather saddle that works for you, it'll probably be a very nice thing; lots of tourists seem to have success with the Brooks B-17.