Foam has a tendency to lose its rebound as it stiffens with age - much like MOST of our parts do. Even if you keep your machine in great condition, the chemicals responsible for elasticity in the foam outgas with the passage of time, weather exposure, and use and the foam's cell walls become more brittle, becoming more plastic and less elastic - retaining more of its butt-compressed form when you get off. What you might be feeling with your 12-13 year old bike is a closer relationship between your tush and the seat pan that didn't exist when the bike was new.
As for compatibility, the assembly should work as the attaching parts and drawings are identical and your seat cover should slip right on - I will defer to those who might have contrary experience, of course. $106.73 at http://www.cheapcycleparts.com/model_years/13430-yamaha-2012--tw200b1/assemblies/257947-seat.
Good luck.
