I just realized that I am one of those players who like to do it the wrong way. But why is it wrong? Apparently, it's better to kill of a 4/2 or any minion that you can kill using Claw alone, without having to use your hero power. The same thing goes for killing any minion with 4 health using Bite. The reasoning behind this is pretty simple: to lessen the risks. But what are you being careful for? Well, it's actually because of the probability that your opponent will be playing Alexstrasza, and of course, your life total might be picked down to 15. So what does this have to do with the hero power of the Druid? Moreover, what does this have to do with armor? The answer is this: Alexstrasza only adjusts your health, but not your armor.
Here's an example:
Situation 1: Your opponent has a Knife Juggler in play. You decide to use Claw and your hero power to have an attack power of 3. You hit the Juggler, and you lose your three armor. Your opponent then plays Alexstrasza a few turns later and you have 15 total health.
Situation 2: Your opponent has a Knife Juggler in play. You decide to use Claw and then attack the Juggler with your 2 attack power. This causes you to lose 1 life point. After attacking, you use your hero power to get 1 armor. So, you lose 1 life point but you gain 1 armor. A few turns later, your opponent uses Alexstrasza. In this situation, you get 15 health and 1 armor.