First of all, let us clear up one thing. Computer science is no more about computers than biology is about microscopes. A computer is a tool that is used by computer scientists to understand mathematical constructs and data structures, and how they behave. Computer science is the analysis of how computing environments behave, how algorithms work, and how to mathematically analyze these systems. It is a science, not an engineering discipline, though computer engineers may use computer science.
Mechanical engineering is an engineering discipline that involves the physical world, how machines work, how materials have various properties, and how to assemble such materials into a physical machine. Mechanical engineers learn about Math, Physics, Material Science, and Chemistry, and the mix of these may depend upon the exact area of mechanical engineering you enter into. I work with a mechanical engineer who is primarily into materials, so he studies more chemistry than physics
I would say that other than mathematics there is little in common between them so that the difference is quite a lot.