Shelly,
Some greenhouses are built for very cold climates and definitely can last throughout the winter. I would not call them cheaper than a cottage. They are quite expensive actually. For instance, a greenhouse roughly 10 x 12 suitable for a cold winter may cost as much as $10,000. A cottage shed from Home Depot is far less, depending on what you buy $1000 - 3500, however they may not be able to withstand a severe cold.
Look up greenhouses using Google.com and you'll see many are built to have many feet of snow on them without a problem. Remember, if you are in a cold climate, a heat light is not going to be sufficient. It will need to have oil or hot water heaters installed in it. It will likely need shades of some sort for when the sun hits it directly or in the summer. And it needs a good foundation to avoid ground freeze.
Paul