I am pretty sure that a "B77W" is a Boeing 777-300 and a "777" is a Boeing 777-200.

The 777-300 is a longer/larger aircraft (the seat map section will give you a better idea) - access to seat selection via manage my booking will show if it's a 3 or 4 class 777-200, you can always cancel out without selecting a seat.

Hope this helps
Bat 21