Additive manufacturing and 3D printing both refer to the process that creates a part in layers—one layer is built on top of another layer. The layers bind together either inherently through the heat of the process or by some chemical means.
Thus, the two terms are often used interchangeably, especially in the media or in reports. 3D printing most often refers to a class of devices known as low-cost desktop 3D printers. These devices are more for home use, hobbies, schools, and so on, and will usually be used to print low quantities of objects—in the range of 1 to 10. The technology they most often use is extrusion or a type of vat photopolymerization (often referred to as stereolithography).
In the media, additive manufacturing is becoming the preferred term to use when referring to floor-sized (as opposed to desktop size) professional machines. (Although a few will still use the term 3D printing when referring to these larger systems.) ASTM F2792 says that additive manufacturing is the official industry term for this technology.
These systems can also use extrusion as the printing technology but they also use technologies known as binder jetting, PolyJetting, other forms of material jetting, any of the laser sintering technologies, electron beam melting, and a host of new technologies being developed. These systems can build one-off objects but they can also be used to build quantities of objects—usually up to 10,000 pieces.