In music, why do they have cut-time when they can just write the music in common time?
The difference is how the music is felt when played. Common time is 4/4 time, meaning 4-beats to a measure with a 1/4 note getting one beat. Cut time is 2/2 time, half of common time, meaning 2-beats to a measure with a 1/2 note getting one beat. Each time specified, has a different sound or cadence, because the first beat of a measure is emphasized. In 4/4 time the first beat is emphasized with minior emphasis on the third beat, EG, DA ta Da ta, DA ta Da ta, etc. But in cut time only the fist beat is emphasized, EG, DA ta, DA ta etc. Tap the different beats out like a drum beat and the difference can be felt. Cut time feels more like marching and so marches are written for cut time. Just guessing, but it appears that cut time looks like common time on the music sheet because of ease of writing, better appearance and ease of reading.