There are a lot of creative ways to increase the difficulty of a creature in a positive way. Only increasing their stats simply makes things harder, not more interesting or fun. I don't think a game should ever be about being harder solely for the sake of it. Dark Souls, for example, was not meant to be such a brutal game. It simply came out that way as a product of the design.
As Scouting Ninja mentioned, diablo gives the stronger enemies special abilities. In Diablo 3, these abilities were randomised. They were all part of a central pool and when an elite monster spawned it would select a number of abilities from that pool that you had to deal with.
Another way to increase difficulty is to improve the enemies AI.
Or you might change its role from 'stronger enemy' to 'group commander'. The elite monster of a pack of goblins is able to command those goblins as an army, rather then having them fight you haphazardly.
Or even change their fighting style. A goblin may wield a small spear to strike at its target. An elite goblin may wield a large spear, using it expertly to attack from a far range, perhaps incorporating more attack variations to mess with the players rhythm.
There are endless possibilities to make an enemy harder. I always notice when a game simply increases an enemies stats and pretends the enemy is now 'harder', and I always hate it. Trust me, it is the perfect example of anti-fun and lazy design. Do not do it.