Jokes aside, the OP ask about weight increase between 1200 GSA and 1250 GSA. The difference,official numbers says it's up from 263 to 268.
The weight increase is due to heavier cam system (shift cam weights more) and the beefed up camchain.
As for the difference between GS and GSA: If you strip down both bikes they are basically the same bike. Slightly different spring travel and slightly different steering angle makes no difference on the scale.
The GSA has a larger, aluminum tank vs the GS plastic tank. GSA has a larger windscreen, and it comes stock with crash bars and luggage rack for alu.-cases. Since 80% full fueltank is part of the given weight, the 50% larger fuel capacity of the GSA ads to the scale. Also, the alu. rims weights less than the spoked wheels of the GSA.