First, there's the brutal tyranny of the rocket equation[1]. Not only must a rocket burn fuel to lift it's payload into orbit, it must also burn fuel to lift all the fuel it will burn on the way plus the rocket parts to contain that fuel and the engines to burn it. With a gun, you only have to put in enough energy to accelerate the payload and capsule to escape velocity. This is far, far more energy efficient! Also, many alternative energy forms are possible to use in such a gun, including electricity.
So what? What about that brutal acceleration problem? Although complex electronics, etc. can be built to survive such stresses, it's a huge design requirement for payloads that might never face such forces again once in orbit or beyond. However, this method of launch is ideal for getting raw materials into space for use in orbital fabrication. Getting big things into space is massively expensive because rockets are so horribly inefficient. If we fire the raw materials up via space-gun and then use them to fabricate things in space, those things could become orders of magnitude cheaper. Mining materials in space is obviously the way we eventually want to go, but that's going to take a while. Space-guns could be a crucial stepping stone.
The Babylon project, had it worked, might not have been terribly useful for Iraq in the 1980's. However, if space fabrication takes off before space elevators are a thing, kinetic launchers will likely have their day.
[1]https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation