Cosmic rays consist of high-energy photons (gamma rays - normally outside the solar system or even extragalactic), electrons and protons (normally from the solar wind except for the very energetic ones).

These photons, electrons and protons collide with particles in the atmosphere and produce cascades of other particles by converting the kinetic energy of the incoming ray into the products. The cascades happen because the first products are energetic enough to, themselves, cause similar collisions further down the atmosphere. The cascades are seen by flying or land-based detectors.

At the LHC you have two high-energy proton beams colliding head-on. There is a cascade but it is a lot cleaner because there is nothing else for the products to collide with, so they just fly off and decay and the collection of decay products is seen by the detector array.

The LHC is a proton-proton collider. When a cosmic-ray proton hits a hydrogen aton in a water molecule you have essentially the same proton-proton collision.

We have met the enemy, and he is us — Pogo