For more information about proverbs, consult the Oxford Dictionary of Proverbs.
From the earliest times until today, these short and pithy sentences have offered advice or pointed a moral. Sometimes, as with Absence makes the heart grow fonder, they express a general truth in an abstract statement. At other times, everyday experience is used to make the point: Don't put all your eggs in one basket warns strongly of the possible outcome. There are also groups of proverbs relating to health (After dinner rest a while, after supper walk a mile), the weather (Red sky at night, shepherd's delight; red sky in the morning, shepherd's warning), and the seasons (April showers bring forth May flowers).
A quotation is thought of as something originally relating to a specific event and person: said by a particular person at a particular time. A proverb is universal, reflecting a general truth: it could be expressed by a number of people at different periods. There is some overlap - proverbs such as Hope springs eternal and Fools rush in where angels fear to tread originated as English quotations - but we think of them as proverbs because they seem to express a general truth, and because the origins of the quotation are no longer popularly remembered.
Every country and language has its own stock of proverbs, and proverbs in our language today reflect every age and time. Art is long and life is short is found originally as a saying of the Greek physician Hippocrates; The apple never falls far from the tree, which means that family characteristics will always assert themselves, is apparently of eastern origin. Sometimes the proverb as we have it today looks back to an earlier period; the idea that Bad money drives out good, recorded from the early 20th century, looks back to the anxieties of the 16th-century financier Sir Thomas Gresham about the debasement of the coinage. The Bible has always been a major source (The leopard does not change his spots!), but changes in the world around us create new proverbs to reflect current experience. The computing world has given us two of the most durable: Garbage in, garbage out and What you see is what you get.
Misunderstandings can arise over the years! Feed a cold and starve a fever was probably intended to represent two separate pieces of advice, but it has been interpreted to mean that if someone with a cold eats too much they are likely to develop a fever, which will have to be starved. When we warn against risking loss or failure because of unwillingness to spend a trivial amount with the words Do not spoil the ship for a ha'porth of tar, we may not realize that the proverb originally ran, Do not spoil the sheep.... The literal sense was against refusal to sanction expenditure on the small amount (or halfpennyworth) of tar needed to protect sores or wounds on sheep from flies.
For more information about proverbs, consult the Oxford Dictionary of Proverbs.