The difference between the two Names is explained succinctly by Ibn al-Qayyim (d. 751):
The reason why the author began the matn with the Basmalah is because he is imitating the example set by Allāh in His Book, in which each Sūrah (chapter) begins with it, except for Sūrah al-Tawbah. The Prophets of the past also began their letters with the Basmalah. The proof from the Qurʾān is the letter of Sulaymān, may Allāh’s praise and protection be upon him, which he sent to a ruler and which began with the Basmalah:
The author is also imitating the practice of the Prophet ﷺ in beginning his letters with the Basmalah:
When mentioning the virtues of the Basmalah, some scholars cite the following ḥadīth which is weak and thus carries no probative force:
Salafi Knowledge 