I don't understand the central-control difference between 1. e4 and 1. e3 The books/articles tell me that the center is d4, d5, e4, and e5.

1. e4 controls d5, but 1. e3 controls d4.

Both control one square in the center, no?

Yet 1. e4 is apparently a vast improvement over 1. e3. 

How would occupying e4 make a difference? Surely a pawn doesn't control the square it's on but rather the squares on which it can capture?

I really wish these books/articles aimed at amateurs would be more precisely formulated. There's just way too much frustrating handwaving. Surely I'm not the only amateur who's had this confusion.
There isn't that much difference in strength between having a pawn on e4 or one on e3 (usually accompanied by one on d4); there are pros and cons to both (and a pawn on e3 can still go to e4, but not the other way around). The difference is between putting them there on move 1.

Let's take a look at the effects of 1.e4 and 1.e3, concentrating on the actual position. For 1.e4:


1.e4 influences d5 and f5, squares on which black might want to place a pawn. 1.e4 d5 2.exd5 gives white chances for an advantage, and 1.e4 f5?? is basically impossible.
If you want to play an opening with a pawn on e4, now is the time to play it there. It's a pawn move to an undefended square, black can easily prevent it otherwise. Observe that the most common replies to 1.d4 are 1...Nf6, 1...d5 and 1...f5, all of which prevent 2.e4.
The bishop on f1 is opened up, and it might go to the nice square c4 (in case of 1.e4 e5 2.Bc4).


For 1.e3:


1.e3 influences d4 and f4, but so what? If white wants to put a pawn there he could already do so, and black can't even legally put anything on those squares next move. So that's not relevant right now.
There is no way in the next ten moves or so that black will be available to prevent a move like e2-e3; it can be postponed until the perfect moment.
White blocks the bishop on c1. Without knowing anything whatsoever about black's setup, white basically already rules out putting the bishop on e3, f4 or g5. If you're going to fianchetto it, why not just start 1.b3. You've just made a concession before black has even played a single move!
The bishop on f1 is opened up, but it can't do much. 1.e3 e5 2.Bc4 just runs into 2...d5 (not enough control over that square).


It's fine to play an opening that involves e3 (as long as you put one or two other pawns near the center instead, otherwise you won't control anything on his half, white is supposed to be attacking), but doing so on move 1 has no advantages. That's a big difference with a pawn on e4, which can be played there on move 1 "for free" and usually only with difficulty after that.