Yes, it is different. Two things happen: the dissolved oxygen boils out, and whatever mineral solids are in there become concentrated as steam evaporates.
Hotter water leads to more caffeine release and a more bitter flavor as it cooks the leaves. If you're serious about the taste of tea, set up four cups and pour water into them: The first boiling, the next after 30 seconds, and on down. Use a cracker between each sip; the later teas should taste slightly lighter and sweeter, and the middle two especially should have a distinct delicate green tea taste. For loose-leaf you normally use a slightly lower temp, while typical teabags need more coaxing to get the flavor out. There are websites that actually list perfect temperatures and steep times for each individual variety, but it's also a matter of taste.
Green tea snobbery can be a little like wine snobbery, the sky's the limit for how sublime you want to go, but at the same time anyone can drink and enjoy it.
Black tea on the other hand doesn't have such delicate taste because it's already pre-cooked. (Oolong retains a little of each nature.) How hot you should make it depends only on how much you like the taste of Bergamot (for Earl Grey) or whatever additives are in your tea, how bitter you can stand it, and how much caffeine you want out of it. The hotter and more bitter it is, the more the tea will cover up the flavor of any additives. Steeping time affects bitterness as well, of course. I'm not entirely certain, but I believe that bergamot oil will also start to evaporate if it's boiled, but I assume you don't boil your tea.
I have no idea how it would change the taste of chai, as I've never made a really good one myself. I typically make them from the powders, and I've seen no difference at all in taste between hot, warm, or even cold water, though the texture changes slightly - it doesn't mix perfectly in cold water. Those powders probably have most of the variability processed out of them. I assume this goes double for hot chocolate, since most of the mixes don't even have real cocoa anymore.
Aside from the possibility of boiling off oils or partially burning green leaves into black, I don't believe the water can get hot enough to change the chemistry of the drinks.
Best Answer
Bear in mind that I'm using an electric kettle, rather than a stove-top one.
First, the advantage of a kettle is that it is quite efficient, and turns itself off once the water is boiling. This as opposed to the microwave, which only stops after a set time, rather than relying on the condition of the water.
Second, a microwave can cause water to superheat, particularly in a ceramic or glass container. This means that there is a certain risk that the water will "jump out" at you once you add something to it or put a spoon in.
Additionally, most kettles have more power than the average microwave, so the kettle is probably faster to boil water.