I ride an inexpensive Trek modified mountain bike and I spend some of my ride on the road and some on the canal, which of course is rather flat. Sometimes at the mid point of my ride I will decide to get off the canal and get onto road with lots of hills. Obviously the mountain bike is not made for this however I enjoy that challenge. Right now I am looking at the Crux Elite EVO. Would that be a good choice, is it worth the money and how much faster would I be on it? Or road bike???
I've had a hardtail mountain bike for a few years, and used it for all kinds of rides -- singletrack, rail trails, roads. A few months ago I got a cyclocross bike (Norco Threshold) in addition to the mountain bike. So I'm in pretty much exactly the situation you describe.
I am enjoying the cyclocross bike and I feel like it's made me about 3-4mph faster. I am surprised that I like the drop handlebars better (more comfortable) than the straight handlebars on the mountain bike.
I already had a set of slick tires for the mountain bike that I installed for road rides, and switched back to knobby tires for singletrack rides. If I only had knobby tires, the speed difference would probably be even greater.
The Specialized Crux that you're considering seems like a pretty good choice. It has 38mm tires out of the box, which is nice especially since you're used to mountain bike tires. Tires of that size have a much less harsh ride than thinner tires (28mm), and don't slow you down (people used to think that, but research has shown that it's not the case).
Two things to consider about that bike: The brakes are cable/hydraulic hybrids; there is a brake cable that connects to a hydraulic piston. I'm not sure if those are as good as regular hydraulic brakes; there is probably a bit of friction in the cable. The other thing is the gearing: for hills you might want an 11-32T cassette instead of the 11-28T. Depends on how strong you are and how steep the hills are.
If you care about riding in the winter, go to a Specialized dealer and ask if you can install fenders on that bike, and what size tires will work with fenders. To attach fenders, you generally need eyelets (threaded holes in the frame). If the frame doesn't have any of those, it makes it very difficult to mount fenders.
The easiest way to tell is by going to a bike shop and test riding few bikes. I had the same decision to make recently (road or CX), and after test riding, I was confident I wanted to CX. The road bike was noticeably faster, but the comfort on the CX trumped that for me, and it was still very fast (33C road tires, compact gearing). It is also running without problems on trails and gravel(as its supposed to), which is a plus as I can take tours through the woods ( as long as the trails relatively flat, eg for hikers/hourses).
I think the main things to consider are: do you care for off-pavement rides? Is the pavement quality good? if there are too many bumps, road bikes can be a pain, while the thicker CX tyres will help.
Finally, depending on what you prefer, you might want to ask to test ride road bikes with a relaxed geometry, that will again be more comfortable.
Also look at touring bikes. Road and CX bikes are made for racing. Which means they have a quite aggressive fit which makes you lean over the handlebars quite a bit. Touring bikes are much more laid back and comfortable, while still being very efficient. They have room for racks, fenders, and big tires which can be nice for riding in the city. If the bike show you're dealing with focuses on the Specialized brand, then take a look at the Awol. If you are happy with Trek, then look at the 520.