I think 'for the sake of' is more serious and shows you are doing something for an important reason. Kind of shows that if you don't do this thing, then there will be consequences. e.g. 'I am earning money for the sake of my family', or 'I am working hard for the sake of my future'. 'For' is more casual and can be used normally, like 'I'm buying some snacks for my friend'. Hope that helps! :)

のために・に　と同じ感じですよ～ Not much of a difference, but "for" is more generic and is used in other situations: - This is fun for me. - This is not for you to decide. - For the last time, I said NO! etc.

Hi Tura, for the sake of someone highlights the fact that you are doing or saying something to help or bring advantage to someone. (i.e. Please do it, for Mary's sake or I hope he's there for John's sake). You could use "for" as well, but the sentence wouldn't get the same emphasis. For the sake of something underlines the purpose, the reason why you are doing or saying something. A good synonym could be "because of". I hope I have been helpful.

Not much difference between them. (After looking in my dictionary) 'for the sake of' means 1) the purpose of getting something - to speak slowly for the sake of clarity, and 2) interest, account - it is done for your sake, it is for the sake of your health.