My sister says there’s no real difference between a child getting the measles vaccine and just catching measles, since either way the child ends up immune. Is she right?

I wouldn’t describe it as “just catching measles.” In the United States, about 1 in 5 unvaccinated people who get measles is hospitalized, and 1 to 3 of every 1,000 children infected will die, according to the Centers for Disease Control and Prevention.