날(day) vs 일(day)

Q. What's the difference between '날' and '일'? Both of them mean 'day', right?

A. Yes, but both have different usage.
 
   '날'is a Korean word whereas '일(日)' is a Sino-Korean word.
   So, '날' are combined original Korean words.

   '날', can be used alone and it has a lot of meanings like whole day, daytime, weather, date, period of time, etc.

  So you can use '날' after nouns, adjectives, and verbs.
  For example,

  나는 비가 오는 날을 안 좋아해.
  I don't like rainy days.

  On the other hand, '일(日)' is used to read dates. It is a kind of counting unit. 
  So, it is often used after numbers like '3일 (동안)', or '(12월) 25일'.

  Also, it is combined with Sino-Korean words.

 For example,
 생일(birthday), 휴일(holiday), 요일(day of the week), 기념일(anniversary) and so no.

  Oh, there is one more usage.
  It goes like this,

  일 3회 복용
  Taking medicine three times a day.

  일 2회 배송
  Items are shipped twice a day.