Gary-I started out the same way as you. I started with a Trek Mtn bike and basically put 800 miles on it riding on the roads. At about 300 miles of the 800, I converted the knobbies to "inverts" and picked up about 10-15% in speed. Thus, assuming the same energy exertion that shows you how much difference tires can make.
At 800 miles I went to my first road bike and picked up another 10% gain in speed over the inverts. I ride in an area with hills and rollers so I'm thinking the last improvement was due to reduced overall weight of the bike, improved aerodynamics plus going from platform to clipless pedals (cycling efficiency). There was also some improvement in the gearing of the road bike over the mountain bike-especially in the big chainring.
When I bought the road bike, I got an adjustable stem so my riding position was not too drastically different than the mountain bike.
Like others have mentioned, you can get a road bike fitted to accomodate your back issue. Based on my experience, I think you'd be really happy with a road bike after you've had some time on it.
Best of luck with your choices. There are lots of options for sure!