Esoteric describes something or someone has being difficult to comprehend, that the number of people who actually understand the concept is limited to a special few. The adverb form is esoterically.
Archaic is a term for things or people that are ancient, and sometimes ancient and not of pertinence anymore. Some synonyms include out-dated or obsolete. The adverb form is archaically.
Note here that each half of the definition can be interchangeable. Archaic items can be old but useful, old and not useful, or just not useful. For the last, it is said with tongue in cheek. The item may be new but already out-dated and so is archaic in its usefulness. And so some items may be esoteric because they are archaic, but not necessarily all the time.
An esoteric explanation: Inside the chrysalis the caterpillar structures are broken down chemically and the adult’s new structures are formed. [Christianity Today]
It’s not surprising that ASP Koh lapses so easily into the esoteric details of crime and the Penal Code – this is a man who has dreamed of being a policeman since he was a child. [Yahoo Singapore News]
I should esoterically note that milkweed, on its own, is a highly essential and functional part of our flora and fauna landscape. [The Sandpaper]
One of the world’s fastest growing airlines with a modern fleet,Qatar airlines has been rocked by reports revealing its shockingly archaic set of rules for female flight attendants. [Pulse Nigeria]
Our current approach to family homelessness stems from archaic beliefs dating back centuries to England’s Poor Laws which regarded the poor as lacking moral development or character when in fact they are simply lacking cash. [The New York Times]
Of course, the Seahawks benefited from Packers head coach Mike McCarthy and his rusty, archaically conservative approach that was accurately blasted by Grantland’s Bill Barnwell. [Bleacher Report]