As a native Californian who has lived in New York for over 15 years, I can say that it is a very big change between the two places. Have you had an opportunity to visit the area in California where you will be located? There is a very big difference, for example, between Northern and Southern California. I think in making this decision, you should at least have the opportunity for a quick visit. How is the school system in the town where you will be? What types of support services do they have? These will be very important in making sure you have a good transition.
I grew up in Riverside, CA. People in New York always think of the wonderful warm weather, and they don't remember that it can be 105 degrees for weeks on end with smog in the summer. My sister-in-law, who grew up in New Jersey, actually went to college in Riverside and couldn't take the heat. Northern California will be much more moderate in climate, so this is why I think it is important to really understand and research what part of CA you will be in.
Finally, what do you and your kids like to do? If your daughters are big ice skaters, I can tell you there won't be a lot of ice skating opportunities in Southern California. If your family is big on winter activities, you may have less of an opportunity to do these things. On the other hand, if you have water babies who love the pool in the summer, they will love to have year round access to open air pools.
The family issue is also hard. It is hard for me to drag my husband and 2 kids across the country to see my family in CA. We try to get there as much as possible, but it is difficult economically on both sides. Right now I am fortunate to have my mother-in-law nearby.
Anyway, I think you should trust your heart, and if you really want to move to Cali, do it, but be sure you research exactly what your community will be like first. Do not make a spur of the moment decision based on the fact that it is really cold in February.