It is very simple, all the season carry different weathers and by setting it to one month later the season changes meaning that the weather the season was carrying has gone. But for example when you are in summer and you switch to autumn the chances are very high that it will be raining.
Seasons with weather effect
- Summer (neutral)
- Spring ( chance of light rain)
- Autumn ( chance of rain)
- Winter ( chances of hail)
However it isn't recommend to change your time in black/white since it may cost some data to be deleted or edited. Also battling Cynthia may become a little harder and she might recite the text she usually recites after battle making her unavailable for battle in some days in Spring