Which is better to say when informing about a future action: "Bi-idhnillah (by Allah's permission)" or "In sha' Allah (if Allah wills)," whether in worldly or religious matters, or is there a detailed explanation for their usage?

There is no difference between making an exception with "in sha' Allah" (God willing) or "bi'idhnillah" (by God's permission), due to the closeness in meaning of the two phrases. Both relate to the attributes of Lordship, and both phrases appear together in Quranic verses with similar meanings. Furthermore, exegetes, scholars of creed, and commentators on Hadith (definition) do not differentiate between them; rather, some have interpreted "mashia" (will) as "idhnillah" (by God's permission). Jurists consider "bi'idhnillah" to be an exception similar to "in sha' Allah."