The verbs look and see have a range of meanings, and the difference between the primary senses in which they are used can be illustrated by examining a common secondary meaning in each case. When you say that you are "looking for" an object, you mean that you do not see it. You are searching for it, casting your eyes around in the places where you think it might be.
Conversely, when someone makes an argument or gives an explanation and you reply "I see," you mean that you have grasped the point. Literal vision is not involved, and you might be looking at anything or at nothing.
These two secondary meanings illuminate the primary definitions of look and see and the difference between them. To look is to cast your eyes in a particular direction. When you are walking on the beach, your friend might say "Look at that!" In response, you then turn your eyes in the direction she has indicated. You do not know if you are supposed to be looking at a ship, or a seagull, or a tidal wave, but you are looking, searching for an object. When you see the object, it is like understanding the point of an argument: you finally see the thing itself.
This is a particularly meaningful distinction when looking at art, particularly in a gallery or museum. After a while, many people find that they can look at a painting or sculpture for quite some time without really seeing anything. This is a form of visual fatigue, analogous to allowing your eyes to pass over a sentence without taking in the meaning.