Difference between "Warm regards" and "Best regards"
Solution 1:
Regards is a synonym for greetings. Either phrase would be used in a formal letter; you wouldn't use these greetings in a letter to a close friend or family member. Warm sounds a bit warmer than best, but basically the two are interchangeable.
"Warmest regards" I use for close friends and relatives.
Solution 2:
I wouldn't use "regards" in any form for close friends or family - it sounds way too formal. However, "regards" on its own in an email is fine for someone I don't know, yet I am having an informal conversation with e.g., admin for the house, kids etc. It sounds friendly, yet not too personal.
Best to use "Yours sincerely" for initial correspondence with business people. "Yours faithfully" is a bit old school.
Solution 3:
Regards = Neutral; Sincerely = Expressing sincerity; Best Regards = Acquaintances; All the best - Professional; and Very Truly Yours = Corporate America
Solution 4:
Best regards seems very positive. Much like saying "Good Luck" or "Have a Great Day".
Warm regards can be taken a few ways. But if I wrote Warm Regards it would mean (to me) "Thinking about You". It sounds very personal.