Interested in Language
what's the relation (if any)
I don't believe there is any relation except thay are all short phrases. This website has boh "Phrasal verbs" and "Idioms" in the menu on the left hand side. In looking at their list of "Idioms" I see several that I would consider "Proverbs" - 'A chain is no stronger than its weakest link.' So I guess there is some confusion of terms among the native speakers also.