As a caveat -
1. All of the following is merely my opinion. I've been diving drysuits since 1986.
2. There are different types of drysuits. Some are made of vulcanized rubber. Some crushed neoprene. Others trilaminate (trilam). Some stretch. Most don't. Its difficult to generalize to all of them.
3. Drysuits are frequently worn with some sort of thermal layer beneath them. What type of undergarment you wear beneath the drysuit often has much to do with your experience of how flexible the suit feels. So, a "drysuit" is a combination of the 'inner layer' and the 'outer layer'.
Now on to your questions:
(lets refer here to a trilaminate or 'shell' drysuit with thinsulate undergarment, which is pretty common...)
Does a dry suit make you feel less 'restrained' than a wetsuit?
Yes, generally speaking. You inflate a drysuit just a LITTLE to offset the feeling of being 'restrained' as you descend. As with all good things in life, however, there is a trade-off. The more you inflate the drysuit (to a degree), the less 'restrained' you feel. However, having a lot of gas inside the drysuit plays havoc with your bouyancy and trim, because when you alter your position in the water column the gas shifts to the highest point of the suit. For this reason most drysuit divers minimize the amount of gas in the drysuit. With no gas at all in the drysuit it can become pretty darned 'restrained' indeed...
What I do mind is the feeling of being "compressed" by the wetsuit (7mm). Does a (properly fitting) drysuit eliminate that factor?
Yes it does. But reconsider the above. It definitely takes some getting used to, to find the perfect balance for you.
Does diving become more enjoyable? Does the freedom of movement increase?
While its a subjective answer, it is certainly "Yes" for myself and for a great many other drysuit divers, I suspect. That being said, recognize that its an environmentally-driven sensation. If I were in Palau, I'd far rather be diving wet. At 200' on a wreck in extremely cold, dark water off Vancouver, I'd be far more comfortable dry. It's not all one way or the other.
What are the down sides (besides getting used to it) to diving dry?
There are a number of examples. Here are three common ones. First, for both men and women, relieving oneself inside a drysuit is problematic. Divers often wear drysuits because they intend to be submerged in cold water for long periods, thus the urge is pretty likely. The answer for men often is a catheter hooked to an 'overboard discharge valve' or P-valve. For women it is often Depends or a similar product. Second, drysuits are notoriously intolerant of neglect or abuse. They demand a more disciplined approach to care, cleaning, and maintenance of the zipper and seals. Third, if a drysuit neck seal or wrist seal lets go on Day 2 of a week-long remote Liveaboard trip, that diver has problems unless they have the spare seals, the glue, and the ability to replace their own seals. It only takes 3 or 4 hours for the glue to set up, but if you haven't got the spare seals, the glue and the know-how, you're likely poop out of luck. A wetsuit diver could simply use a bit of adhesive and be back in business in 15 minutes.
So drysuits require a more demanding and disciplined approach, as well as some skills that wetsuit divers don't need.
That said, for more demanding diving (cold, deep water, etc.) they are the only way to go.
Just my opinion, your mileage may vary.
Doc