what is the difference?
The Marine Corps is the primary expedition force of the UNSC. They are usually stationed on ships to be first responders to a threat or act as a deterrent force on a newly colonised planet. While the Army is a more established force primarily serving to provide ground side planetary defence to more industrialized and important colonies.
During the war, the Marine Corps saw the brunt of the fighting as they would planet hop from one battle to a next along with the Navy, while the Army would attempt to further fortify a planet against invasion alongside the Air Force. Both forces were very successful in engaging the Covenant on the ground, usually besting numerically superior enemy numbers with tactics and equipment like the Grizzly, but once the orbital battle was lost they could do very little to defend. In the halo lore there is the 53rd Armoured Division, they are a perfect example of what I explained prior.