They both seem to have translation as "fundamental truth" or "cardinal principle" except 原理 seems to be much more common than 妙諦. Does anyone have some information on the nuance of these two or how they are used differently in a sentence?
Kenkyusha’s 新和英中辞典 translates “原理” as “a principle, a tenet and a law,“ but it doesn’t provides the translation of "妙諦," perhaps because of the relative low incidences of its usage.
大辞林 published by Sanseido defines "妙諦" as "優れた真理 - the superlative truth,” and 新明解国語辞典 published by Sanseido defines it as "そのものの存在として高く評価できる良さ - excellence that can be highly evaluated as a being on its own," which I think is a bit obscure.
Thus broadly speaking, “原理” refers to a principle, and "妙諦" refers to truth.
We say “そこに禅の（武道の、茶道の、書道の）妙諦がある - There is the ultimate truth (or goal, tenet, secret) of Zen (or Japanese martial art, tea ceremony, caligraphy).”
"妙諦" is often used for the meaning of "the essence of teaching" like ”禅の妙諦," but it isn’t always connected with Buddhism. It can be applied to anything such as studies, arts, sports, games, techniques, and the way of life that you challenge.
原理 refers to a basic principle or credo that should be shared by everyone as the starting point. In religional context, it's something that if you don't believe this you wouldn't be a Christian, Muslim, Buddhist and so on. There's a word 原理主義 that means fundamentalism.
妙諦 is more like essence in the sense of "what is essential is invisible to the eye", that is, something is not easy to recognize and only attained after pursuit, practice, reflection etc.
In my opinion, 妙諦 feels like a principle of Buddhism, whereas 原理 is more like scientific fact. Sometimes 原理 can also mean theorem.