The reason for the discrepancy is because your ancestor originally enlisted in Company H, 2nd Colorado Infantry. That regiment consolidated with the 3rd Colorado Infantry in October 1863, thus forming the 2nd Colorado Cavalry, which was the unit your ancestor belonged to at the time of his discharge (accounting for that latter regiment being listed on the Certificate of Service).

Your ancestor shows up on the rolls of Company H, 2nd Colorado Infantry under the name of Joseph H. Flinn. I bet if you contact the National Archives and order the military file for him using that spelling, and listing that regiment, you'll have a whole new avenue of research open up to you. The Archives doesn't compile the different military files for a single individual who served in multiple units, so sometimes you have to do a little sleuthing to come up with them all.