What is the difference between "into" and "onto"?

As reported by the NOAD, into has the following meanings:

1. expressing movement or action with the result that someone or something becomes enclosed or surrounded by something else. (Cover the bowl and put it into the fridge.)
2. expressing movement or action with the result that someone or something makes physical contact with something else. (He crashed into a parked car.)
3. indicating a route by which someone or something may arrive at a particular destination. (The narrow road that led down into the village.)
4. indicating the direction toward which someone or something is turned when confronting something else. (with the wind blowing into your face)
5. indicating an object of attention or interest. (a clearer insight into what is involved)
6. expressing a change of state. (a peaceful protest which turned into a violent confrontation)
7. expressing the result of an action. (They forced the club into a humiliating and expensive special general meeting.)
8. expressing division. (three into twelve equals four)
9. informal (of a person) taking a lively and active interest in something. (He's into surfing.)

onto means:

1. moving to a location on the surface of something. (They went up onto the ridge.)
2. moving aboard (a public conveyance) with the intention of traveling in it. (We got onto the train.)

As side note, in some cases it is correct to write onto, but in some cases it is correct to write on to.

You climbed onto the roof.
Let's go on to the next chapter.

kiamlaluno's answer is comprehensive. A simplified answer is

put A into B

means A is somehow inside B and enclosed by it.

put A onto B

means B has a horizontal surface on top of which A now rests.

It's the difference between in and on.

You get into a cockpit and onto a highway. We go in to be able to say we were here but we are always the ones who move on to the next one.

Inside the cockpit, on top of the highway. We go in(side) (in order) to be able to say we did it; when you are ready to move on to something better let me know; I'll get you onto the waiting list.