Vision works with light rays. For example, in your eye, light is coming from the top of your monitor, being limited by passing through a hole at the front of your eye called the pupil, going through your eye's lens for focus, then continuing in mostly the same direction and hitting the the bottom of your retina, the back wall which is covered in light-sensitive cells. In the same way, light comes from the bottom of your monitor and hits the top of your retina, comes from the left and hits the right, and comes from the right and hits the left. (Your brain then flips it around!)
This idea of rays of light from multiple locations in a scene all coming to a point, crossing, and continuing on beyond the hole is important for all cameras.
In a pinhole camera, the rays are directed through a tiny hole and hit a photographic film, changing it chemically. This film can then be used to create a picture (remember: flipped just like in your eye so it will appear correctly).
Camera obscura was an important invention as it lead to the development of the traditional camera. It uses mostly the same setup, except rather than letting the rays hit film, they're directed against a wall or screen for direct viewing (and often tracing) by people. Sometimes mirrors are used between the pinhole and the screen to flip the image to appear correctly. Most camera obscura setups were large enough people would be inside the camera!
I hope I did an okay job explaining them. In a nutshell, the only absolute difference between the two is whether or not the camera uses film! (Though small camera obscuras were rare, and large pinhole cameras were rare).