Yep, exactly what the title says.
If they are different, then what differs between AP and OBEs?
Some people describe obes and astral projection as the same thing. Others say that Astral projection is its own thing… And others say that astral protection is not real but is just a dream.
All I can say is that all these people are experiencing something, make up your mind.
It seems to me that astral projection = out of body experience = dream
Just my thoughts and ideas 
Yes and no.
Comparing OBE’s and AP is analogous to comparing rectangles and squares.
An out of body experience is exactly that; an experience. Having an ‘experience’ of something doesn’t necessarily mean what you are describing it as is true. That is to say, an OBE doesn’t have to be actually outside of your body, so long as it looks, feels, etc, as if it is.
If you disagree, please try to think of another experience. Say a near death experience. Say someone sets up an elaborate prank that results in you nearly being hit by a truck. The truck truthfully may only be a hologram and incapable of hurting you, but if you don’t know the difference, is it really a different experience?
Now the other side is astral projection. AP can be defined as “projecting your consciousness onto the astral plane and traveling about.” This inherently tends to imply that you /are/ leaving your body. So yes, an astral projection would be one example of an OBE.
AP, however, relies on a bit of difficult to prove theories. If you believe what you experienced to be an AP, then so be it. If you think it was just a dream, ok. In either case, it would not be false to call it an OBE, but in both cases it would be untrue to call it the other case. (That is, calling an AP a dream or calling a dream an AP)
[edit]To amend my last statement. That is my belief, though it is not an uncommon belief that one can travel through the astral plane via dreaming. In such case, it would be both a dream and a projection.
Dream is an astral projection. Not vice versa. Your soul travels every night, and when you wake up it is shown to you as a dream because it would be too much for our brain. That’s why it’s easier to astral project when you already did it. Because your brain can take it. OBE is something really wide. It can be a dream, it can be an astral projection, it can be a near death experience, psychedelic experience etc,etc, but it is not considered an OBE if you’re not aware of it in the moment (don’t know why, because we are out of our bodies every night, whether we know it or not). This was explained very good in a book, I can’t remember the name.
There are interconnections with all these experiences… Read about the pineal gland and such things. You wouldn’t believe it, but a coctail of psychedelic drugs is resposible for all these experiences. Why these chemicals are in our brain is still a big question.