Is it permissible to say that Allah does what He wills, or that Allah does everything He wills, and what is the difference between the two?

Both statements are permissible to say. However, it is better to say: "What Allah wills, comes to pass, and what He does not will, does not come to pass," because it stems from belief in Allah's universal, preordained will. Will (إرادة) in the Book of Allah is of two types: a universal, preordained will, and a religious, legislative will. The universal, preordained will must inevitably occur, whether Allah loves it or dislikes it. This aligns with our saying: "What Allah wills, comes to pass, and what He does not will, does not come to pass." As for the religious, legislative will, it only pertains to what Allah loves, and it may or may not occur. If the intention behind "will" is the universal, preordained will, then there is no difference between the two expressions. If the intention is the religious, legislative will, then there is no difference in the case of a believer.