If they were in the military the same time, why would they be different ranks?

It was established that Frank and Hawkeye were drafted, but not necessarily at the same time. The max time soldiers served in the Korean War was 18 months. After that, regular Army were sent to a different unit and draftees were sent home. So let's say Frank was drafted at the beginning of the war and Hawkeye was drafted a few months later. That would be a possible reason why Frank was promoted first. Plus Hawkeye and Trapper were constantly getting into trouble, even more than Frank. There's no way Hawk and Trap were getting promoted with all the stuff that they pulled.