What difference would using Rock Sugar make in tea? I like my tea sweet, and generally add several cubes of white sugar. Teavana sells rock sugar, which it claims is somehow "better" while being vague on what that means. Is it sweeter? Less sweet? Somehow interacts with the tea leaves differently? 

Interesting, you're meant to brew it with the leaves, presumably sweetening the whole pot rather than doing it to taste for individuals, which sounds annoying. Is that just marketing or does that have some benefit?
In Northern Germany, sugar crystals similar to the style depicted on the linked web page are used for "East Frisian Tea". East Frisian Tea is more a style of tea service than a specific variety of tea, but usually some blend of Indian black teas is used, often Assam and Darjeeling, but some variation from that is possible; the main thing is that it's not an aromatized or fruity tea, but a black tea blend, and tends to be infused with the intention of being slightly bitter.

When serving, rock sugar is placed at the bottom of the cup. Tea is poured over the rock sugar. Then, typically, but not always, a bit of cream is poured on top of that. You don't stir what's in your cup, so you start with a creamy flavor, progress toward a fairly bitter/astringent black tea, and have an increasingly sweet finish. You keep on repeating the process of adding sugar, tea and cream and drinking until you've run out of things to talk about with your friends.

The slow melting process is the main desirable attribute of this type of sugar, not the flavor difference. The brown crystals, with a slight molasses hint, were the most common ones I saw in Germany, but clear ones were not unheard of either. The hint of molasses is the only flavor difference from the brown crystals.