There are LOTS of reasons I think they will make the world a better place but I’m just going to focus on the two I’ve done research in.
The first is finding new ways to treat cancer by using particle beams rather than x-rays. The particle beams are much more accurate and don’t damage as much healthy tissue as x-rays, meaning that cancers can be treated better and with less side effects. (This already exists! Check out Charged Particle Therapy or Hadrontherapy)

Another one which I’m currently working on is using an accelerator to drive a completely safe type of nuclear reactor. The idea is that on it’s own, the nuclear core (which in this case would contain Thorium, rather than Uranium) doesn’t fission on it’s own – it needs a very high power source of neutrons to set off the reaction. The accelerator I’m currently designing make a beam of very high power protons which then go on to make neutrons to run the reactor. If anything goes wrong, the accelerator shuts off and the reaction stops, so it’s much safer!
Some of the other benefits are that there’s more Thorium on Earth than Uranium, so we have a much longer supply of it, and that there isn’t anything nasty like Plutonium created, so the “wrong” people can’t make bombs out of the waste. On another note, it would produce a LOT less waste than normal reactors and could even be used to burn up existing radioactive waste! I think it’s a winner all round! (The accelerator design is challenging, but I’m working on that!) If you want to learn more, it’s called an Accelerator Driven Subcritical Reactor, but was first called an Energy Amplifier by Nobel Laureate Carlo Rubbia (who came up with the idea!)