The first appliance that most people will not have in Japan is a clothes dryer. Japan hangs almost all of their laundry so get used to having socks, shirts and underwear hanging around your apartment at all times of the year. I know some people in the USA who hung their laundry but I had never done it before moving to Japan.
When it rains in Japan many people will take their wet clothes to a local laundromat and dry them there. Also Japanese washing machines are usually quite small so you can only wash about half of what you would in a washing machine in North America. Large blankets and things of that nature you will most likely have to take it to a laundromat to wash it.
Japan does sell dryers though. Most people just don't have the space for them or don't want to pay the electrical fees. You can get a decent dryer for around 300 to 400 dollars.
Slightly related subject but most people in Japan never wash their futons. They will wash the cover but they will never wash the actual futon. Most people just bring it out onto their porch and smack it with something that looks like a tennis racket and somehow that makes it clean.
The price for something like this will run you around 300 dollars. You can get a used one for really cheap but do you really want something used other people used to wash their dirty clothes in?
They can cost you upwards and over 1000 US dollars. Not planning on moving for a while? This might be for you.
The cheapest you are going to find them for is around 250 US dollars and the cheapest ones are so small that it makes buying them almost pointless. Do you want to make 6 cookies at a time?
Even in my worst apartment ever in the US I had a heater built into the unit that worked off natural gas that was provided by pacific gas and electric company. In Japan most of the time your heater will not be built into your apartment or house. You'll have to buy a heater and either hang it on the wall or connect it to a huge kerosene tank that you keep somewhere inside your house (no I'm not joking).
I am not sure about many other countries but in the USA you have natural gas coming into your apartment from gas lines that run under the city. In Japan there is nothing like that to my knowledge. The gas company comes and replaces your gas tanks once a month and measures how much you used and that is how you get billed. These will most likely be outside your apartment. In one of my apartments I had the entire complex's gas tanks right next to where I slept and next to the parking lot too. So, if anyone ever crashed into them I would surely have been dead.
If this didn't already sound terrible enough, if you live in Hokkaido or Tohoku your heating and possibly even your water heater will be running off kerosene. I have no idea why this is and why the heater can't use normal natural gas just like my heater back in the US did but that's the way it is. These kerosene tanks are huge and they cost a lot of money to fill up so having a little extra cash no standby is a good idea if you live up north. Banks close early and don't operate on the weekends very much and running out of kerosene could possibly kill you.
Air conditioning is not that big of a deal. If you are in a place that is hot enough to need air conditioning than buying one and installing it shouldn't be that big of a deal. They work very similarly to an air conditioner in the US so not a lot to talk about there.