These two phrases seem to be interchangeable in most cases.
But I found one case where it seems that "in order to" works, and "for the sake of" sounds like it's not as good a choice of words.
- I've probably continued to sing in order to write these words.
- I've probably continued to sing for the sake of writing these words.
To me, in order to seems to be a better choice. For the sake of doesn't seem to be as good a choice, because it doesn't seem to be a good choice in general for things that you're going to do in the future. For the sake of seems to be more appropriate for things that already exist, not actions that are going to take place in the future. For actions that are going to take place in the future, in order to sounds sort of more appropriate.
Am I right? Is there anything to what I'm saying?