Generally speaking, a coach is closed and a carriage is open.
Think of a stagecoach. It’s closed. There are doors, and a seat up front for the driver. Or, think of a coach of state like the royals ride in from Buckingham Palace to Westminster or to Parliament. Closed.
Then, think about the carriages in Central Park, NYC. They’re open, with a bench for the driver. No doors, no roof, nothing. Open.
The words aren’t readily interchangeable, regardless of the Wikipedia article about them.