What the LHC does is slam hadrons -- large collections of quarks bound together by strong nuclear forces -- into other hadrons at high energy. The LHC uses the hadrons it does not because there is anything special about them but because it's somewhat easier to get the energies they want to study using the hadrons they choose. They also chose the energies they use for the collision for convenience more than anything special. Ideally, they want the most energetic range they can accurately control. If they could build a bigger collider, capable of higher energy collisions, they would, but these things are complicated, big, and expensive.
Cosmic rays are a mixture of fast particles, including hadrons of various sizes, traveling at very high speeds. Many cosmic rays are bare protons, the same as used in the LHC. The energy range of cosmic rays is wide, ranging to many more orders of magnitude higher than the LHC. A collision between a proton from space at 100TeV and a proton in an oxygen atom in the upper atmosphere of the earth is very similar to a proton-proton collision in the LHC, but much higher energy.
If I am interpreting a graph on Wikipedia correctly, cosmic rays with an energy of over 1000 TeV impact the Earth at a rate of about 1 per square meter per year. Given the size of the Earth, that's 14 million/second. So 14 million collisions hundreds of times more energetic than the LHC can do happen in the Earth's atmosphere every second. And there appears to be a power scaling going on. 10TeV cosmic rays are thousands of times more frequent than 1000TeV cosmic rays.
The difference, and why the LHC was built, is location. Looking at cosmic ray collisions tells us what the end result is going to be, but it doesn't tell us what happens partway through. If you look at a car crash on the side of the road, you know that the car got squished and the driver was injured. If you look at a car crash in a lab with cameras and crash dummies, you can tell that the driver hits the windshield before the crumplezones absorb all the energy.
The same sort of thing with the LHC. If the LHC will create Higgs Bosons, they are being created all the time in the upper atmosphere. But Higgs Bosons are expected to last an incredibly short amount of time, and all we see is what's left after they decay into other particles. We can't see cosmic ray collisions clearly enough to see if the decay particles come from Higgs or from other processes we understand well.