Soyuz is 1960's tech that doesn't have much payload capability. The shuttle was 1970's tech that was a terrible compromise when all the other projects that surrounded the Apollo replacement program were cancelled.
The shuttle was supposed to be a taxicab that literally "shuttled" crews to a space station that was supposed to have been launched by Nova or some other heavy-lift vehicle. As originally designed, the shuttle had a small or no payload bay, the entire launch system was to be 100% reusable, and was supposed to be a vehicle that could launch again 1 week after returning from space. It was supposed to be essentially an X-20 Dyna Soar, that was launched by a larger space-plane type vehicle.
But all these plans fell apart when Nixon cancelled everything except the shuttle. The Air Force stepped in to "save" the program, but at the cost of a small, cheap, re-usable vehicle, and they added the large payload bay for their military projects, because they needed something to lift all those spy satellites. Gone was the re-usable, piloted launch vehicle, and what we ended up with was... well, whatever scraps were left.
Soyuz was a design with purpose, but a very limited purpose. The shuttle we got was a wide-mission-scope boondoggle that was trying to be all things to various branches of government AND be a NASA space vehicle. That it was able to fulfill even most of the roles it was tasked with is a miracle when you consider what a hack job it was versus what it was meant to be.
This is not even an Apples to Oranges comparison. It's an Apples to Chicken comparison.