“Best regards” and “Warm regards” are two phrases commonly used in modern written English to end a letter or other message. They are both formal, and will almost never be encountered in common conversation or most social media contexts. “Best regards” is usually used between acquaintances, general well-wishers and solicitors. It is considered to be emotionally neutral and rather polite. Regards is a positive word, with connotations of respect and well wishing, and English speakers always enjoy the superlative. “Best regards” is considered somewhat officious and emotionally neutral.
“Warm regards” is a rarer phrase, used only in very specific circumstances. It has connotations of heat and physical touch. It is only used between friends or people who know each other in social circumstances, and it carries with it a friendly and slightly unctuous sense of wishing to get to know the other better. It is also used in expressions of sympathy. It is sometimes found in letters between distant relatives, especially the older generations. It would be considered slightly odd to end business letters with “Warm regards,” unless the people involved knew each other on the social level. One would generally not use “Warm regards” unless they had met on at least one social occasion and knew the other person well enough to inquire after their health their family.
“Best regards” should be used for people who do not know each other, were as “Warm regards” is mostly used specifically with social acquaintances. In an expression of sympathy, such as a letter of condolence after the death of a relative, it would be more acceptable to say “Warm regards” or even “Warmest regards.”