Saying, n.
That which is said; a declaration; a statement, especially a proverbial one; an aphorism; a proverb. Many are the sayings of the wise, In ancient and in modern books enrolled. Milton.
Syn. -- Declaration; speech; adage; maxim; aphorism; apothegm; saw; proverb; byword.
Shades of meaning
Fernald has no entry of its own for saying; he lists it under proverb.
Where he writes about it:
A saying is impersonal, current among the common people, deriving its authority from its manifest truth or good sense; as, it is an old saying, "the more haste, the worse speed." A saw is a saying that is old, but somewhat worn and tiresome. — from proverb
A byword is a phrase or saying used reproachfully or contemptuously. — from proverb