What is the difference between a 民國 and a 共和国?
The term 'Republic' originated from Latin 'res'(entity, concern) and 'publicus'(of the people, public)

'民國' and '共和國' both means 'republic' 


'民國' emphasize on the meaning of 'people' (民)
'共和國' emphasize on the meaning of 'public' (共)


China for most of its history, was an empire.  All the power belongs to the imperial court. The people(民) had no power at all, therefore 'republic' was  translated as '共和國', emphasized on the word '共'(public)- '共和'(public together)- '共和國'(nation of public together)

When Sun Yat Se and his party 國民黨 defeated the Qing Dynasty, ended the imperial rule, they chose to name the new China '中華民國'(Chinese republic/ republic of China) to emphasize China was no longer an 帝國 (empire). 

Another reason to choose the name '中華民國' was the word '民' can also represent '民族' as in '中華民族' 

Nowadays, most republic in the world are translated as '共和國' in Chinese, because not all republic is made up of mostly one single 民族 like China(中華民族) and Korea (大韓民族).