What is the difference between "Indeed, God does what He intends" and "Indeed, God does what He wills" in Surat Al-Hajj, and what is the secret behind choosing "intention" in the first and "will" in the second?

As Ibn Ashour mentioned, al-iradah (will) and al-mashee'ah (desire/wish) are synonymous in meaning. This is supported by the statements of exegetes regarding the verses: "Indeed, Allah does what He wills" (إِنَّ اللَّهَ يَفْعَلُ مَا يُرِيدُ) and "Indeed, Allah does what He desires" (إِنَّ اللَّهَ يَفْعَلُ مَا يَشَاءُ). Both verses have been interpreted to mean that Allah does in His creation what He desires, whether it be reward or punishment, honor or humiliation, and He is not questioned about what He does. Despite subtle linguistic differences between them, al-iradah and al-mashee'ah are cosmologically identical in this context.

As for the difference between al-iradah ash-shar'iyyah (legislative will) and al-iradah al-kawniyyah (cosmic will, which is al-mashee'ah): the legislative will only pertains to what Allah loves, while the cosmic will encompasses both what is loved and what is not. The use of both phrases in the noble verses is, and Allah knows best, due to elaboration and stylistic variation to emphasize Allah's iradah and mashee'ah.