Why draw blood from veins rather than arteries?
Veins have several advantages over arteries. From a purely practical standpoint, veins are easier to access due to their superficial location compared to the arteries which are located deeper under the skin. They have thinner walls (much less smooth muscle surrounding them) than arteries, and have less innervation, so piercing them with a needle requires less force and doesn't hurt as much. Venous pressure is also lower than arterial pressure, so there is less of a chance of blood seeping back out through the puncture point before it heals. Because of their thinner walls, veins tend to be larger than the corresponding artery in the area, so they hold more blood, making collection easier and faster. 

Finally, it is somewhat safer if a small embolism (bubble in the blood) is introduced into a vein rather than an artery. Blood flow in veins always goes to larger and larger vessels, so there is very little chance of a vessel being blocked by the embolism before the bubble reaches the heart/lungs and is hopefully destroyed. Blood flow in an artery, on the other hand, always moves into smaller and smaller vessels, eventually ending in capilllaries, and there is a chance that a bubble introduced by a blood draw (generally rare) or more commonly an intravenous line (IV) could block a small blood vessel, potentially leading to hypoxia in the affected tissues.