Best is to look at definitions of the words and their use in context. Those are definitions from Merriam Webster:

apothegm: a short, pithy, and instructive saying or formulation: <a href="https://www.merriam-webster.com/dictionary/aphorism" target="_blank" style="color: rgb(38, 86, 103);">APHORISM</a>

-> her mother endlessly repeated the <em style="color: rgb(34, 95, 115);">apothegm</em> “cleanliness is next to godliness”

adage: : a saying often in metaphorical form that typically embodies a common observation 

She reminded him of the <em style="color: rgb(34, 95, 115);">adage</em>: "A penny saved is a penny earned.

2: a <a href="https://www.merriam-webster.com/dictionary/terse" target="_blank" style="color: rgb(38, 86, 103);">terse</a> formulation of a truth or sentiment : <a href="https://www.merriam-webster.com/dictionary/adage" target="_blank" style="color: rgb(38, 86, 103);">ADAGE</a>

the high-minded <em style="color: rgb(34, 95, 115);">aphorism</em>, "Let us value the quality of life, not the quantity"

So, you see that there are only slight nuances between those words.

Maybe these links might be helpful:

<a href="https://www.dailywritingtips.com/proverb-vs-adage/" target="_blank">https://www.dailywritingtips.com/proverb-vs-adage/</a>;

<a href="https://www.dailywritingtips.com/aphorisms-mottos-and-sayings/" target="_blank">https://www.dailywritingtips.com/aphorisms-mottos-and-sayings/</a>;

Aphoristic collections, sometimes known as wisdom literature, have a prominent place in the canons of several ancient societies, such as the Sutra literature of India, the Biblical Ecclesiastes, Islamic hadiths, the golden verses of Pythagoras, Hesiod's Works and Days, the Delphic maxims, and Epictetus' Handbook. 

Adagia (singular adagium) is the title of an annotated collection of Greek and Latin proverbs, compiled during the Renaissance by Dutch humanist Desiderius Erasmus Roterodamus. In literature one can often read: The collection of aphorisms by Erasmus are called 'Adagia'. So the differences are often subtle and overlapping. Adagia are considered a sub-species of aphorism such as epigram.

Apothegm and adagia are not really used in everyday language. Proverb and idiom are used most commonly. Aphorism is used in English and as well in German quite often, but rather in academic circles.

firstly, are you planning on using these in every day speach? Because Aphorism and apophthegm are like, never used in every day speach. I had to look them up to know what you were talking about.

Proverb is listed twice... so thats already one less on the list :)

Idiom, adage and proverb are probably used the most in spoken English (With adage and idiom most frequently used)

So... are you using these in everyday conversation? Or are you writting a thesis? :)

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Same with me Jacque, I honestly never even heard of apothegm or aphorism. Proverbs and idioms to me are just 'sayings'. Adage... I've heard of that word... though I'd need to look it up.