Choosing between MA and MSc in International Relations

Is there any practical difference between getting an Masters of Arts in International relations and getting an Masters of Science in International relations?

Will the focus be any different? Will the job prospects and career paths afterwards be any different?

Different schools, which are on relatively the same level in terms of prestige and quality, have the same major, but different schools have the degree as an MA and others have it as an MSc. Should this be a factor at all in deciding which to attend?

No. Look at the course catalog/curriculum, the program’s career services and internship/job placement records, and the professors and special programs and resources they have. But the letters themselves don’t matter.