The Pick axe is the tool that you use to mine ores and other things like Stone or Obsidian. There are made out of three items (wood planks, cobblestone, iron ingots, gold ingots and diamonds.
Each one is faster at mining and stronger than the one before it (except gold which is weaker than Iron and stone). It can be used to mine Emeralds, Obsidian, Stone, Redstone, Diamonds, Iron Ore, Gold Ore, Coal and Lapix Lazuli. It can be enchanted to be more efficient, stronger or produce double when it mines an ore,
The Axe is the best tool in Minecraft for getting wood as it is faster at getting it. Like all the other tools it comes in Diamond, wood, Iron, Gold and stone form and can be enchanted to be more efficient and to be stronger. It can be used on all types of trees (Dark Wood, Spruce, Oak, Birch and Redwood).