The term "premium motor spirit" is more commonly used in some countries outside of the United States, while "gasoline" is predominantly used in the US and a few other regions. Both terms refer to the same substance with virtually no significant difference in composition or application.
So, to answer your question, yes, gasoline and premium motor spirit are essentially the same thing. It refers to the same refined petroleum product that powers automotive engines, and any distinction in naming is mainly regional or country-specific.