They can be synonymous, but that is not always the case. "Around the world" means all the way around back to the starting position, as in the entire planet. "Across the world" suggests a great distance, as on the opposite side of the planet.

My thoughts: Around the world refers to going from one point to the next throughout the world eg. I’d love to travel around the world. Whereas across the world mostly refers to going from point a to b on either ends of the world eg. My dad flew across the world to see me perform. Hope I helped 😊

Flying around the world suggests you're coming back to the starting positon, but flying across the world suggests one direction, to the other side of the world. Hope that helps 🙂🙃

To me it’s the difference between the fixed points;going around the world starting from a one specific point and coming back to the fixed one.While going across doesn’t necessarily point to one fixed direction

Not a whole lot, in my opinion.