"All The Best" vs "Best of Luck"
Solution 1:
As a native American English speaker, I don't know if there is much of a difference. At the very least, no one has ever taken me to task for using either one with the inappropriate age group.
In fact, they both seem to be ellipses of the phrase I wish you all the best of luck, it just depends on where you cut the phrase.
I wish you (all the best of) luck.
I wish you all the best (of luck).
Solution 2:
All the best is more a farewell gesture than best of luck; the former has an air of finality that the latter lacks. If you say "all the best" to someone, there's an expectation that a significant amount of time will pass before you meet one another again. Best of luck doesn't have such an implication, I believe; if you were to go to Vegas, and bet your life savings at the craps table on a 30:1 shot, I might say to you from sidelines, "best of luck", though I readily expect to be there comforting you in just a few moments hence. Saying "all the best" there would sound very strange, if you knew we weren't going to part for some time.
Solution 3:
I would say that "best of luck" would refer to something more specific, Whereas "All the best" is a generic well-wishing.
- Best of luck in the new job
- Best of luck with your exams
- All the best for the future
etc.
So if you knew someone was having a job interview you might say "Best of luck for tomorrow!" but if you said "All the best" it would sound more like you were wishing them well for a long time - so you might say it if you didn't know when (or if) you were going to see them again. It's more an expression that you hope all goes well in their life.
I've also seen "All the best" used as a way of signing off in an email or letter (instead of "Sincerely" or "Best regards"), although it is not overly common. It would be strange to use "best of luck" here.
Solution 4:
British English: 'All the best' is used interchangeably with 'Best regards' or 'Best wishes' or even 'Take Care' or 'Best for now' or 'Bye for now' when signing off a letter or email. It's friendlier than 'Best regards' or 'Best wishes' but not as familiar as 'Take Care'.
'Best of luck' is entirely different and specific to a need to wish someone luck because they are doing something involving some small or large risk.
Solution 5:
I'm a British person who uses a lot of little terms and expressions probably without thinking too much about them I suppose. I use the term, "all the best", to shorten the phrase, all the best for the future, where "best of luck" is another thing altogether and just as short already.