The weather on Fuji is everything. Fuji is hot, cold (-5c to -15c on the summit), windy, rainy, and sunny; sometimes all in the same day. You need to be prepared for it all because you'll probably get a little of everything over the course of your climb.
Generally August is the driest and hottest month for climbing, while early July and late September to October gets quite cool.
If you're thinking about what gear/equipment to bring, when making this decision ask yourself, "if I walked outside in a rainy downpour for 6 hours in these shoes/jackets/pants, would I be happy or miserable?" Generally, this is a good guide. Check out our required equipment list for the 2-day Mt. Fuji tour for a comprehensive guide on what to bring with you on your climb.