Yep, cornet and flugel horn are quite different instruments. Same bore size though, I believe. The 'plumbing' is tubed around in the cornet.
Much of the difference is a result of whether the instrument is cylindrical (straight bore) or conical (increasing bore). The trumpet is mostly a cylindrical instrument, the flugelhorn is mostly conical and the cornet falls in between.
The trombone? Obviously a comical instrument.