Rapid Shot is a full round action. You get an extra shot at your highest BAB and take a -2 penalty. It is roughly equal two2 weapon fighting (the first)in that sense.

Manyshot is a standard action. You fire a number of arrows depending on your BAB and take a larger penalty. Manyshot should only be used in a situation where you want to move and then fire (or vice versa) because you take a much larger penalty and only the main arrow can deal precision damage. (Critical Hit, Sneak Attack, Skirmish etc.)