What is the difference between swearing by Allah and taking an oath by Allah, and what is the expiation for swearing by Allah to commit a sin, such as masturbation?

Oath and swearing are synonymous; both are a form of yamin (oath). There is no difference between swearing by Allah and making an oath by Him. Their expiation is the same if the swearer breaks the oath, and it is the expiation for an oath mentioned in the words of Allah the Almighty: "So its expiation is the feeding of ten poor people from the average of that which you feed your [own] families or clothing them or freeing a slave. But whoever finds not [the means] - then a fast of three days. That is the expiation for your oaths when you have sworn." It is not permissible to swear an oath to commit a sin. The swearer must abandon it and atone for their oath.