I tend to use the words script and alphabet fairly interchangeably – I might talk about the Arabic script or the Arabic alphabet, for example. However I just noticed today that Wikipedia has one page on the Cyrillic script, which focuses mainly on the history of the script/alphabet, and separate pages for Cyrillic alphabets as used for particular languages (Russian, Bulgarian, Serbian, etc). The Latin/Roman script is treated in a similar way.
It seems that a script refers to the overall writing system rather than to the particular set of letters used to write a particular language, or alphabet. So you could talk about the Devanagari script, and the Hindi alphabet, i.e. the Devanagari letters used to write Hindi.
Have you come across this distinction before? Do you think it’s a useful one to make?