Open is basically another age group after senior, it's probably easiest to think of it that way, though the age ranges differ between the 3 open divisions. Open 4, Open 5, and Open 6.
International means that it's a worlds eligible open division. I think there might have been a distinction between Open and International before, but they're basically the same thing as
luv2cheer92 said. People usually refer to Open 5 and International 5 to distinguish it from Senior 5.
Since other countries don't follow our age grid, they go off of the International designation since for many of them their "senior" teams are older than our senior teams, so they compete in the International divisions, which, to them, aren't really open teams. If that makes any sense.
Open 4 is NOT international since it's not a worlds division, it's also only a trial division for now.
And I've actually seen competitions separate Open 6 and Open International 6 for those going for worlds bids and those not, which is just weird.