The comments here are generally calling these “old-style” figures, but—while I’m aware that this is what some people call them—I think the term is a poor one. The old-style figures are better referred to as text figures and the so-called “usual” ones—which @funkah prefers—as lining or titling figures.
The difference between them is not really simply a stylistic choice: text figures are basically lowercase numbers and titling figures are uppercase ones. Saying you don’t like them is like saying you don’t like lowercase letters, and that you prefer everything in uppercase. Using titling figures in running text is like typing in all caps—it’s screaming at you.
It’s not so much that the text figures are old and were abandoned, but rather that typewriters simply left them off to simplify things. And since computer keyboards were based on typewriters, they never made their way to computers either. And then they started being used exclusively in newspapers and advertising. If you look at books, though, you’ll see that they never really went away. Only now are we finally getting them back—and that’s the entire point of OpenType and it’s a good thing.
Now, the difference between text and titling figures isn’t precisely the same as upper- and lowercase, since titling figures are generally used in titles despite the fact that titles can be mixed case; and they’re also used in tables of numbers. (Although there is another dimension of figure variation—namely, whether they are proportional- or fixed-width—and fixed-width lining figures are typically used in tables so that the places line up.)