Everything on the record would limit the number of people able and willing to come forward. Not everyone is savvy or quick witted, and the knowledge that not everything they say will be used is of great relief to them. Also, full disclosure is freeing, rather like confession, and so telling all can be very appealing to someone who is relating a story, even if they could not in good conscience allow parts of that story to become public. Moreover, just because a person is relating a story, it does not make them a writer. It can be very helpful for a reporter to have the thoughts and motivations behind actions, even if those thoughts are too personal to print directly.
Lastly, while it’s true that reporters are professionals doing a job, it would not be fair to pretend they are no longer human when they are on the job. Newsworthy tales are often emotional and can have very powerful and very personal meaning to the ones who relate those tales. Going off the record allows a source to speak with another human being, just one person to another, without exposing potentially fresh and vulnerable wounds to the entire world. Often times, some of the appeal of telling a reporter your story, is simply being able to tell your story.
"Off the record" is important. It is a matter of trust, and trust should be a part of every profession, especially reporting.
While it is true that Power should have known better as a result of her position, it is also true that a good reporter would have clarified the matter before using the material. Not doing so was irresponsible, and it could be consider unprofessional, immoral, or both.