What is the difference between saying: "In the the name of Allah, the Most Gracious, the Most Merciful" and saying: "By Allah, the Most Gracious, the Most Merciful"?

There is no significant difference in meaning between "Bismillahir Rahmanir Rahim" (In the name of Allah, the Most Gracious, the Most Merciful) and "Billahir Rahmanir Rahim" (By Allah, the Most Gracious, the Most Merciful), as both denote seeking assistance from Allah. However, "Bismillah" is more eloquent and comprehensive, and it suffices for all other expressions of seeking assistance, whereas "Billah" necessitates seeking assistance with only that specific phrase. It must be noted that the Lawgiver commanded saying "Bismillah" in certain instances, such as prayer and some invocations, and saying "Billah" is not sufficient in those cases.