The difference between "I really need a holiday" and "I definitely need a holiday" lies in emphasis and nuance:
"I really need a holiday" – Emphasizes the intensity or depth of the need. It suggests that you are feeling very tired, stressed, or overwhelmed and strongly feel the need for a break.
"I definitely need a holiday" – Emphasizes certainty. It suggests that there is no doubt in your mind that you need a holiday, possibly based on how you’re feeling or recent events.
Both convey the same general idea, but "really" focuses on how strongly you feel the need, while "definitely" highlights that you are sure about it.