
Take a look at these two images, both the tail end of the first stage of a really big rocket. One is from almost sixty years ago. The other is contemporary. Both have a lot of rocket engines designed to work together. Can you tell which is which?

https://i.sstatic.net/45zNk.jpg

https://spaceflightnow.com/wp-content/uploads/2023/02/20230208raptors.jpg
OK, the graininess of that first image may have given it away. Those 30 engines are on the first stage of the N-1 rocket, the rocket that the Soviet Union hoped would take their cosmonauts to the moon. As we all know, the American moon rocket, the Saturn V, worked spectacularly. The second image is of the SpaceX Super Heavy booster, the first stage of the Starship launch vehicle. It has 33 rocket engines.
The N-1 rocket flew four times between 1969 and 1972, none successfully. The record for the Super Heavy is mixed, but it has scored some notable successes in itself, independent of the success of the Starship second stage.
Why did the N-1 fail and the Saturn V succeed?
There were two ways to build a rocket big enough to take men to the moon in the late 1960s, each with its own problem. You could build really big engines and cluster a few of them together in the first stage, the one with the hard job of lifting the entire vehicle from the launch pad. This was the approach taken with the successful Saturn V. Five enormous engines powered its first stage.

https://www.clarionledger.com/gcdn/-mm-/92a9acee110601543bc673fc67b15afad0432a11/c=0-62-1277-783/local/-/media/2016/06/17/JacksonMS/JacksonMS/636017701705752423-saturn-rocket-stage-2.jpg?width=660&height=373&fit=crop&format=pjpg&auto=webp
The problem with such huge engines is something called combustion instability. In such a large engine, with massive amounts of fuel and oxidizer mixing and igniting, turbulence and pressure variations could multiply and blow the engine apart. The solution lay in the design of the engine’s injector plate. A plate sat at the top of each F-1 engine. Holes across the plate’s surface forced the engine’s propellants (liquid oxygen and kerosene) into the combustion chamber. The instability was solved by the addition of baffles (dividers) across the injector plate’s surface.

http://heroicrelics.org/info/f-1/f-1-injector/f-1-engine-injector.jpg
Static testing (firing the engines on the ground) gave NASA the confidence to fly the rocket. The Saturn V flew 13 times between 1967 and 1973, all successfully.
The other approach is to use a smaller rocket engine and cluster a lot of them. That was the Soviet approach. In the rush to get to the moon first, the Soviets did not run static tests. The problems were not so much with the engine itself; in many ways it was a very efficient design. Without diving too deep into the details of each failure, we can say that the coordination and control of thirty engines was just too much to manage.
But Starship has 33 engines! Why does it work?
Well, it hasn’t always. But it hasn’t always failed, either, as seen in this image of all 33 engines firing.

https://cdn.mos.cms.futurecdn.net/ZW6aFHbMYgrgQ6fe4Emnfn-1200-80.jpg
The shortest answer to the posed question is 21st century computer control versus that available in the late 1960s.
The Saturn V was a marvel, but that first stage used brute force. SpaceX’s Super Heavy Booster, if it can ever get past the testing stage, really is an elegant and efficient rocket.