Matter was designed to solve a market problem: to make it easier for non-technical consumers to select devices that would work with what they already had at home. Which meant one device that could work with either Apple home, Amazon Alexa, Google assistant, or SmartThings. (And also potentially aqara, Tuya, Ikea, and some additional platforms that don’t have their own voice assistants. But each platform, including the big four, have been rolling out matter support bit by bit, with different bits for each platform, which is part of the current confusion.)
Zigbee and Z wave function at the networking level, just like Bluetooth, Wi-Fi, and thread.
Matter is at a level above that, the application layer. So it’s not going head to head in the way you were asking about. Right now, the options are matter over Wi-Fi, matter over thread, or matter over bridge (which can bring in really any protocol that someone wants to build a bridge for, right now, primarily Zigbee or Bluetooth, but also a couple of proprietary protocols).
So its end goal is to make it easy for people to buy and set up devices. That’s really it.
Unfortunately, during the first year in 2023 it’s spectacularly failed at that goal, because the independent third-party standard organization allowed the use of the matter logo long before things were standardized. 
so it meant more research was required, not less.
Hopefully that will change in the future, we will just have to see.
But in any case, it’s not designed to be an improved protocol. Just an improved buying and setup experience.
It’s true that matter does require local operation, which could be a plus for some platforms like smartthings, but Apple home and a number of Z wave and zigbee platforms already operate mostly locally, so again, it’s not aiming for a technical advantage.
The detailed discussion of matter, and how it impacts smartthings is in the following thread:
Matter - smart home connectivity standard (formerly Project CHIP)