Contrary to popular belief the snow conditions at resorts that are easily reachable by bullet train from Tokyo can be just as good with just as plentiful snow as in Hokkaido. Although if you crave powdery conditions you may prefer the slopes in Hokkaido resorts due to the low moisture level of the snow due to Siberian winds coming across the sea towards northern Japan (it means it can get pretty cold too!).
You can see the daily snowfall on this site below (no affiliation).
http://www.snowjapan.com/e/daily/A_TO_Z ... wDepth.php
If you wait until later in the season or even February onwards, the snow depths can be pretty serious. I remember last season some resorts not far from Tokyo such as in Niigata and Nagano prefectures as well as in Hokkaido were over 5 metres!
If you just want to go skiing for the whole time then you may find it better to go to Hokkaido where you will get amazing powder snow conditions and probably more English spoken due to the higher numbers of Aussie/Kiwi ski/boarders who also own a lot of winter holiday homes there.
If you have never been to Japan before you could combine the ski trip with a few days in Tokyo or Kyoto for the cultural side and good shopping in Tokyo (and plenty for the kids to marvel at). It can be expensive to travel by bullet train from Tokyo to the mountains which only takes 1-1.5 hours but what the Japanese do very well is organisation. You can buy a package for a few days or weekend which includes the train, bullet train, hotel, ski pass, ski rental and even ski clothing, gloves etc.
I recommend paying a bit more for a better hotel or traditional style ryokan where dinner may be included in the overnight price and getting a hotel with a good onsen (hot spring pool) is a must. It's the perfect way to relax after a long day blasting the slopes. The outdoor onsen (think they are called rotenburo) are amazing with the snow all around you yet sitting in a really hot pool taking in the sights of the mountains. Will be a truly unforgettable experience. Makes me want to go back there for a bit of snowboarding!
In my experience ski trips in Japan are never cheap and although it's a case of you get what you pay for, the quality of food and service is generally very high so I don't think you can go too far wrong.
Reality is an illusion created through lack of alcohol..