In short, wet hopping refers to the state of the hops being used (still fresh and full of moisture) and dry hopping refers to the point in the process the hops are added (late in the process, in the fermenter when he beer is cold). Dry hopping has nothing to do with the state of the hop itself. In fact all hops used for beer are dried for a very good reason, but let's start at the beginning:
Hops are a seasoning for beer, adding unique flavors and aromas based on the variety of hop used. However, like the herbs you use to season food in the kitchen, hops can be used dried and stored over long periods of time or used just after being freshly picked, at harvest time. However, hops are only harvested once a year: late summer/early fall in the northern hemisphere and in the spring in the southern hemisphere. Since we need to make beer all year long (not just in the fall or spring), most hops are dried immediately after being picked, packaged and stored for later use. Most hops that go in beer are used in this state.
However, we do have the unique opportunity to use freshly picked hops once or twice a year, twice if you can get your hands on some of that southern hemisphere harvest and get it overnighted from afar-far. We make our wet hopped beer just once a year in the fall, getting the hops shipped in from Michigan and using them within 24 hours of when they were picked. It is necessary to use them quickly, because hops in this state (still full of moisture) will degrade and their aromatic qualities will go from awesomely delicious to terribly stinky fast. However it is well worth the effort. Like using fresh vs. dried parsley, the underlying flavors will be similar to the hop variety's dried counterpart but with an added freshness that is hard to describe beyond, "it tastes...green(?)".
Since we use wet hopping to describe the use of freshly picked hops, still full of moisture, you'd think dry hopping would refer to using dried hops. However, almost all hops used in brewing have been dried, and we add these dried hops multiple times throughout the brew process: on the hot side, while the wort is boiling, later in the boil, in the whirlpool as the wort starts to cool and after the wort is cooled and sent to the fermenter, where the yeast turns wort into beer. It is only at this final stage, in the fermenter while the beer is cold, do we refer to adding hops as "dry-hopping". It is strictly refering to the point in the process we add the hops, not the state of the hops themselves. Why do we call this process dry hopping? That's another story and, frankly, I don't know that one.