Well, look at the Quake system.
In Quake, you have 3 different kinds of armor: Green, Yellow, and Red. With Green armor, the armor absorbs 1 point of damage for every point of damage the player takes. In that way, it acts like doubling health. However, consider the complexities of that situation.
If you have 10 health, and you pick up a 100 health powerup, you go back up to full 100 health. However, if you pick up 100 Green armor, you now effectively have... 20 health. Remember: you die when you run out of health, not armor.
Point #1: It can reward you for maintaining a state of high health.
Consider the same situation: you have 10 health. You come across 100 health, and then you come across a 50 health pickup. You still have 100 health. Joy.
However, if you have 10 health and come across 100 Green armor, and later get a 50 health pickup, you now have effectively 120 health.
Point #2: It provides a way of breaking the health cap, while still allowing the cap to exist and matter.
Now look at the Red armor. For every 6 damage you take, 5 goes to the Red armor, and 1 to you. Red armor comes in lots of 200. Even if you are at 10 health, you effectively have 60.
Point #3: Different grades of armor allow for different qualities of effects. Some can provide more of a reward in certain situations, while others are more situational rewards.
Thus leading into:
Point #4: It allows you to provide more rewards for player exploration. It's a different kind of reward from health, weapons, ammo, and other powerups.
Remember: Quake is not a modern-style hyper-linear FPS. It's more of an adventure game where you shoot people. Rewarding exploration is important, so you need a good set of viable powerups of different qualities. Armor is merely another quality to choose from.
This is why modern FPS games don't use it. They don't need it, since they're very linear games.