계속 and 계속해서
Any difference in meaning follows their function in the sentence.
When they describe the manner of something continuing, they are completely interchangeable.
In such usages, 계속 is simply a short form of 계속해서, so there is no difference at all.
- 비가 이틀을 계속/계속해서 내렸다. => It's been raining continually for two days.
- 계속/계속해서 같은 말만 하고 있다. => (He) keeps saying the same thing.
- 사업에 계속/계속해서 실패했다. => (He) kept failing in his business.
Sometimes, ~해서 in 계속해서 means a reason or background. In this case it can't be shortened to 계속.
In such usages, 계속해서 is equivalent to 계속한 끝에, 계속했기 때문에, or 계속하다 보니까, etc.
- 실패를 계속해서 가세가 기울었다. (The subject 가세가 separates 기울었다 from 계속해서)
=> After a string of failures, the family fortune declined.
- 공사를 계속해서 시끄럽다. (Structurally, it is [공사를 계속해서] 시끄럽다. "공사를 [게속해서 시끄럽다]" makes no sense)
=> Continuing construction work makes it noisy.
All the above examples are their adverbial usages, but 계속 is sometimes used as a noun too.
- 이번 회는 지난 주 이야기의 계속이다. (This episode is a continuation of last week's story)