What is the difference between armor and health?

There are some cards that provide armor to a hero. At first armor seems to be the same as additional hitpoint, with the exception that you can go higher than the base 30 health.

In which ways does armor behave differently than simply additional health? What are advantages and disadvantages of armor compared to an identical number of health?

In most cases an extra point of armor will be the same as an extra point of health would, but there are certain cards that have synergies with low or high health, or armor. A few examples of such cards would be:

• Shield Slam - Deals damage to a minion based on the amount of your armor.
• Alexstrasza - Sets a hero's health to 15, leaving armor unaffected.
• Molten Giant - Mana cost drops -1 for each point of health you're missing from your hero's max health (usually 30).

In addition to 3ventic's answer, note that hits fully absorbed by armor don't count towards "on damage" effects such as water elemental.

In addition, your HP is capped at 30. +Armor allows your effective-HP to go above 30, meaning skills like the Warrior's (+2 armor) and Druid's (+1 armor +1 attack) are still useful even at full HP, while the Priest's (+2 HP) is not.