The premium client offers one-click updating of all your add-ons, which is a very nice feature for those of us that have a few hundred of them floating around. When I tried this earlier today the updating went smoothly and without any problems.
Another feature of the premium service is that you're not subjected to all the ads on the Curse website while you're logged into your account. This is good if you're still going to the site often, but I'm not sure how often you'll go to their site if you're using their add-on downloader client.
The other big things that Curse advertises as part of their premium package is that you'll get access to premium only beta key giveaways, however there is no indication as to exactly what these betas are yet.
In line with supporting the Curse site (which you're doing by giving them money for the client), you're also supporting the add-on authors. "A part of your subscription will go to addon authors, helping them develop and maintain the Addons you love," according to the material presented on the Curse website.
Finally, and perhaps the most attractive thing to some, is the promise of faster add-on download speeds via the client. Having used both the regular and this new premium client, I was able to see a slight increase in speed, although it wasn't anything substantial. However with that said, I suspect that this will be a big deal on patch days when everyone is clamoring to update their add-ons.