Is the weather an important influence to our willpower brain and the effects can be seen on the daily work out?

Yes, I believe it is.

I believe most people gain weight during the winter because the gloomy weather outside demotivate them from leaving the comfort of their home. It is also a known fact that humans eat more during winter. Those two factors are bad for anyone trying to get in good shape. 

In my opinion, the correct mindset should be to think of working out and exercising during the winter in order to get the summer body you want!