A proverb is a short, common saying or phrase. It particularly gives advice or shares a universal truth. Synonyms for proverb include byword, which can also be someone or something that is the best example of a group.
Adage is also listed as a common synonym for proverb. Adages tend to be old, known for decades or centuries, and share universal truths.
Since the words are listed in the definitions of each other, they are interchangeable and neither could be called incorrect. If one wants to create a distinction, proverb should be used for saying that give advice and adage for sayings that are particularly old. However, when one is referring to an adage from the book in the Bible titled Proverbs, it is always called a proverb.
Examples
The Chinese have an old proverb for this expressly symbolic and potent silencing act: “Kill the chicken to scare the monkeys.” [Bradenton Herald]
Thus a good number of proverbs suggest that warm or fine weather at the beginning of the year is bad news, especially for the coming harvest. [The Guardian]
Arnold was quick to agree with the old adage that ladder position – especially with the Wanderers sitting last – does not matter in matches such as these. [Sydney Morning Herald]
We thought instantly of surpliced choirs mouthing such adages as “Ending the deficit by 2020, fixing the roof while the sun shines, Tory competence versus Labour chaos” in an exciting new liturgical plainchant. [The Independent]