Although they got it right in the Halloween episode, they say the blizzard occurs Christmas Eve in 1876, but that was a Sunday, so what are the kids doing in school, and would not a teacher then know about the speed at which weather can change in such parts, so why would she send them out ?
Despite it being late December in Minnesota, Charles and Isaiah - as do several other characters - spend the first portion of the episode wearing only shirts, without any winter or cold weather gear whatsoever.
Despite there being 30-40 mile per hour winds (according to the telegraph messages) the snow on the ground isn't seen to blow about. In winds that heavy the air would be filled with blowing snow.
At 43:57 when Charles opens the shutters, the window that was broke by the tree earlier appears not to be broken anymore. For the rest of the episode, the shutters remain open with no sign of the broken window. If it was still broken, it would be way too cold to leave the shutters open for that long amount of time.
After Mary, Laura and Carrie find the shed, there are cracks and slats where you can see the outside. No snow can be seen falling despite scenes to the contrary.
At 3:28, Charles and Isaiah pull up in the wagon outside Jim Bowers house and the wind is blowing very strong because a storm is coming in. In the distance behind the cabin the trees are not blowing in the wind. They are perfectly still.
The shack where Laura Carrie and Mary started the fire was full of hay in reality it would've went up in flames and they also didn't put it out when they left also would've went up in flames.
The term blizzard is derived from the word blaze, and was first used in the USA during a big snowstorm in the winter of 1880-81.