- e4 has an advantage over 1. e3 in that it does not potentially block the bishop on c1 (one the d-pawn is moved) and also that it attacks f5, thus preventing black developing his bishop there (or the knight via e7), and attacks d5 preventing black putting a knight there (after developing it on f6 or e7). It also hampers black moving a pawn to d5 although it doesn't prevent it. (Black is unlikely to want to move the pawn from f7 to f5 at this point).
1 d4 has similar advantages over 1. d3.
Black however will often counter with a more "defensive" move that involves moving a pawn just one square. This will often "support" another pawn advance. The plan will often be to "fianchetto" the blocked bishop, or simply develop it to d7 or e7 in a more defensive role.
Although White, with the extra tempo, could start with such a move and adopt a defensive strategy, it is more frequent for White to want to use the extra move advantage to adopt a more "attacking" strategy. ("You're supposed to be at home".. as we might say in another sport.. however the principle kind-of applies in chess. White more attacking, Black more defensive).
I have been known to start a game 1. c3. Caro-Kann or Slav with a tempo. At least my opponent is less likely to be prepared for it than they are against 1. d4 or 1. e4.
(The only 2 serious tournament games I've played in the last 27 years, I was Black both games so it didn't come up).