~에 비해서 and
~보다 are attached to nouns and used to compare things, such as:
지난 주에 비해서 바쁜 편이에요.
지난 주보다 바쁜 편이에요.
We are somewhat busier compared to last week.
When is one preferred over the other? Is there any difference in meaning?
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These two forms are often used in very similar fashions. Up to personal preference, they can be used in the same types of ways most of the time. Personally, I use 보다 much more frequently ("N+에 비해서"란 말 보다, "N+보다" 더 많이 써요).
From a literal perspective, N+에 비해(서) means "compared to N," whereas N+보다 means ".... than N," as in "more than N," or "smaller than N."
Technically, the form N+에 비해(서) is a postposition (에) combined with a verb (비하다). The form N+보다 is just a postposition.
There are some times where N+에 비해(서) can be used, but N+보다 cannot be. Here are some examples of such cases.
나이에 비해 어른스럽다. He is old for his age.
그 아이는 나이에 비해서 퍽 성숙했다. She is rather mature for her age.
그의 목소리는 뚱뚱한 몸집에 비해 상당한 고음이구나! For such a husky fellow, his voice sure is high!
A slight alteration on the N+비해(서) form is N+비하면, meaning 'if you compare N to/with...."
노력에 비하면 성과가 미비하다. The results are subpar in relation to our efforts.
All of these examples employ the "to compare to/with" meaning of 비하다.