Does an old tire become tight?
Certainly, old tires get stiffer as the rubber dries out, and this makes them harder to remove and install.

But what you've probably experienced is the difference between brands and styles.  Some tires are a hair oversize and slip onto the rim quite easily, while others are a hair undersize and are much more difficult to deal with.  It's hard to predict which you will encounter unless you are familiar with the brands involved.

(The leak you experienced after replacing the tube was probably due to damaging the tube while installing it.  Unless you really have to, DO NOT use screwdrivers to lever the tire off/on, but use plastic "tire irons", or, better, a "Quick Stick" tire lever.  Using screwdrivers frequently causes tube damage.)