The current Falcon rocket always expends a stage and lands a stage. Starship will land both stages. Also Starship is far larger and in rocketry there is something called the square cube law that means bigger is better. So its far more efficient. That leads to far cheaper mass to orbit costs.

In order to land not just the first stage but also the second stage more fuel will have to be carried on the ascent phase just to enable the recovery phase. This then necessitates weight savings elsewhere to avoid payload to orbit being cut way back. SpaceX chose to remove the landing system from both, the first and second stages (both because, after all if you can eliminate the landing system from one stage then you can eliminate it from the other).

This and other weight savings will enable high payload to orbit in a fully reusable launch platform.

Landing without legs requires something like this catch system -- something never done before, and clearly very difficult to do.

For humans to have any reasonable presence somewhere else in our solar system (moon, mars, etc.) we need the ability to launch tons (literally tons) of stuff to orbit and to the destination. and we need to launch it often to do anything in a reasonable amount of time. the only way to do that is to make reusable launch systems. SpaceX's Falcon 9 has aced that for satellites (see Starlink and everything else they've launched). The Starship launch system is capable of launching a significantly larger payload, ~20 times more. What they demonstrated yesterday is that a launch system that is capable of getting us anywhere in the solar system can be reusable. Huge accomplishment.