오랜만 is clearly different from the other two:
It means "It has been a while"
오랜만이에요: It has been a long time (since we last met)"
Both 오래 and 오랫동안 mean "for a long while" , but 오랫동안 is more "heavy", "severe", etc
오래 기다리셨어요?: Have you been waiting for a long while?
오랫동안 기다리셨어요?: Have you been waiting for a very long time (ages, entire life)
As a rule of thumb, if you see two similar words, the longer one is usually the "stronger" one.