For me, "proverb" and "old saying/saying" are interchangeable (同義語), with "saying" being more common. "Maxim" means that the saying sort of has a moral (教訓) to it, and "epigram" means that the saying is kind of sarcastic (皮肉) as well as poetic (詩的). "Maxim" and "epigram" are both rarer words. I would usually only use "saying" or "proverb" unless I was writing something. There's also a bunch of other words that mean very similar things, such as "adage", "aphorism", etc. all of which are also things I'd mostly say in writing.