Gardeners LOVE to complain about plants changing their scientific names. I certainly do it as much as anyone else (I even sing about it) but, though I certainly wish the new names could be easier to spell and pronounce, I actually don't mind it too much. Let me explain.
Scientific classification of organisms changes are we learn more about them. Go far enough back in time, and people classified whales as fish. As we learned more about them, however, that classification became obviously incorrect. Whales may live in the ocean, like fish, but they breath air, are warm blooded, and produce milk. Whales, it is clear, are mammals, like us and dogs and elephants, not fish. So the official classification changed.
For as long as scientists have been classifying things, they've been doing it based primarily on what they look like. Taxonomists peer at tiny details of flower structure and pollen grain shape and decide "I think that's an aster" or "Salvia, for sure." Recently, however, there has been a revolution. We've started looking at plant's DNA.
DNA sequence is the ultimate answer for deciding what is most similar. Two plants may look similar, but look at their DNA, and you can see exactly how different they are.
The result has been chaos in the world of scientific names. The genus Aster has been split into a billion pieces. Your Sedum 'Autumn Joy' is now Hylotelephium. It is frustrating (though, you gotta admit, Hylotelephium is REALLY fun to say!) but there is good news. You can't get any more fundamental than DNA sequence, so once this wave of revisions is over, names should be more stable than they have been in the past.
The other good news is the new names actually accurately reflect plant's relationship to each other. And why should you care about that? Well, let me give you an example. Take Hibiscus syriacus, the hardy, shrubby Rose-of-Sharon. Many gardeners still know it as Althea, which is the genus of mallows. Althea, Hibiscus, who cares? Well, you should. Knowing it is a hibiscus, breeders realize that it is closely related to other species of hibiscus, and started trying to make hybrids. One result is this lovely plant:
Hibiscus 'Tosca' (image from Arrowhead Alpines -- where you can buy it!) This is a hybrid between H. syriacus and H. mutabilis with bigger flowers on a bigger, more vigorous plant! Knowing it as althea would only result in unsuccessful attempts to cross it with a mallow. Better names make for better breeding which means better plants for you in your garden.