Hmm - tricky question. :) Whenever I got confused with two English synonyms, I've always wondered what either of them sounds like to a native speaker's ears. Sometimes their intuition is more informative than dictionaries, so I'm going to say what my intuition tells me. 예 is more polite than 네. 1) 네, 알겠습니다. 2) 예, 알겠습니다. There's nothing wrong with the first - but it's true that the second is more polite, euphonious. This perfectly explains why I instinctively use 예 when I need to be more polite in some situations, like when you're talking to your teachers. Sometimes I say 네 in a particular tone when I'm being sarcastic. Like 네에에~↗ Hope that helps ^^

In most of the Korean grammar books, "yes" is basically translated as 네. On the other hand, people may use 예 in daily life conversations, especially in informal situations, possibly because of ease of pronunication. So, I personally expect that 네 is for more formal cases, while 예 is for more daily life informal cases. Nevertheless, it will be nice if there could be some native speakers to help us confirm (or rectify in case of mistakes) the above presumptions. Good day.

well 네 and 예 is synonyms. But by Korean Standard Language Regulation(표준어 규정), it says '네' is 표준어(Korean Standard Language) and '예' is allowed to use as same meaning in same situation. so it's not like one is formal and the other is informal. Both are very polite expression and words, so you can use either one to express like 'yes, grandmother' '네(예), 할머님(할머니, 조모님)' informal way to say 'yes' could be '응', 'no' could be '아니'. Formal way for 'no' could be '아닙니다(아니요)'. hope it could be helpful to you. have a great nite. :)