Indicates a greater degree between two compared items
When to use
two-item comparison, simple comparison, increase expression
The nuance
더 is used as a comparative to compare two items or states.
더 (deo) · 가장 (ga-jang, gajang)
더 expresses comparative degree between two items, while 가장 indicates the superlative among multiple items.

One lane per word — no nested boxes, just the gist.
Indicates a greater degree between two compared items
When to use
two-item comparison, simple comparison, increase expression
The nuance
더 is used as a comparative to compare two items or states.
Indicates the highest or best among multiple items
When to use
top among many, superlative expression, emphasizing the best
The nuance
가장 is used as a superlative to indicate the top or best among many items.
이 사과는 저 사과보다 더 달다.
i sa-gwa-neun jeo sa-gwa-bo-da deo dal-da., i sa gwa neun jeo sa gwa bo da deo dal da.
Comparing two apples, so '더' as comparative is appropriate.
이 사과는 가장 달다.
i sa-gwa-neun ga-jang dal-da., i sa gwa neun ga jang dal da.
Indicating the sweetest apple among many, so '가장' is correct.
이 사과는 더 최고다.
i sa-gwa-neun deo choe-go-da., i sa gwa neun deo choe go da.
'더' and '최고' both express comparative/superlative, making it redundant and incorrect.
그 영화는 다른 영화보다 더 재미있다.
geu yeong-hwa-neun da-reun yeong-hwa-bo-da deo jae-mi-iss-da., geu yeong hwa neun da reun yeong hwa bo da deo jae mi iss da.
Comparing two movies, '더' correctly expresses comparative.
그 영화는 가장 재미있다.
geu yeong-hwa-neun ga-jang jae-mi-iss-da., geu yeong hwa neun ga jang jae mi iss da.
Indicating the most fun movie among many, so '가장' is appropriate.
그 영화는 더 재미있다 최고다.
geu yeong-hwa-neun deo jae-mi-iss-da choe-go-da., geu yeong hwa neun deo jae mi iss da choe go da.
Combining comparative and superlative in one phrase is unnatural.
나는 더 열심히 공부한다.
na-neun deo yeol-sim-hi gong-bu-han-da., na neun deo yeol sim hi gong bu han da.
Using '더' to express studying harder compared to before or another.
나는 가장 열심히 공부한다.
na-neun ga-jang yeol-sim-hi gong-bu-han-da., na neun ga jang yeol sim hi gong bu han da.
Using '가장' to indicate the person who studies the hardest among many.
Which word indicates a greater degree between two items?
Which word indicates the best among many items?
What is the role of '더' in "이 책이 다른 책보다 더 재미있다."?
What does '가장' mean in "그는 반에서 가장 키가 크다."?