Definitely do some research about the specific areas you want to visit to see what will be open. Green Gables Heritage Place is only open through October. I think the Anne of Green Gables Museum might be open all year, but not sure (it's not, see link posted by E&Cmom.) I have only been in the summer.
ETA: Green Gables is absolutely worth it if you can go in season. I'm a huge Anne fan, and was visiting PEI for family reasons, but made sure to carve out time for this.
For Maine, it really depends on where you're going. I live in a coastal southern Maine town that shuts down in the winter. I think two restaurants stayed open last year (and only a few days a week) and one store stayed open on weekends until Christmas Eve and then closed for the season; the rest closed sometime around Indigenous People's Day. I'm sure the ski and other winter sport destinations might have more available that time of year, but keep in mind some are not easy to get to if the snow is bad already.
Last year, we had a big coastal storm right before Christmas that knocked out power to a of southern Maine and washed out some roads. And this is the milder winter area of the state.
That said, I have spent Christmas in the Camden area and remember quite a bit being open and having a wonderful, but it has been years, so that may have changed (and that trip was visiting family, so most activities were not out and about.)
