I regularly ride MTB off road and used to use the bike for commuting too. I recently bought a hybrid for my commuting in addition to my MTB. What should I consider in terms of maintenance and riding style on the new hybrid. The main differences for me are the skinny tires and rim brakes instead of hydraulic discs.
First, term hybrid is sometimes used for quite different bicycles (rigid or suspended front fork, caliper or canti rim brakes)
I think there are no big difference in maintenance. Maybe to check more often your wheels and spokes since they might be more stressed.
As for riding style, you have to go more easy on curbs, and all kind of holes on the road. Your brakes will be less effective, especially in the rain or snow.
First thing that you have to watch out for, when switching to slimmer tiers is the turning. I used to make very steep turns with my mountain bike. The first day on a touring bicycle, I ended biting the asphalt.
Second you have to be careful with jumping of sidewalks and potholes. Most likely your new bicycle doesn't have a suspension, and excessive tension can result in damage in the rims (especially if they are a cheap model).
Most likely you will have longer breaking distance. It is very important to know how fast you can come to a complete stop, especially if you are riding on the road. Needless to say, cars can break very VERY fast (first hand or should I say "head first" experience). I recommended buying a computer, so that you can measure your speed. I prefer not to exceed 30kmh in urban environment.