Essentially none. 3D printing is a type of additive manufacturing. I suppose it might be possible to come up with some process that isn't 3D printing, but is technically additive manufacturing, but it would be a stretch.
You could make the argument that the word "manufacturing" in additive manufacturing makes AM more of an industrial term and 3D printing less so, but it's a very subtle distinction.
Manufacturing methods can essentially be grouped into the following three categories:
Subtractive - Material is successively removed from a solid block until the desired shape is reached
Machining processes (turning, milling, drilling, etc.)
Formative - Mechanical forces are applied to material to form it into the desired shape
Shaping processes (bending, molding, pressing, etc.)
Additive - Material is manipulated so that successive pieces of it combine to make the desired object
3D-Printing processes (SLA, SLS, FDM, etc.)
The best example of this is interlocking parts: In materials like ABS plastic (Acylonitrile Butadiene Styrene), Polyamide, Alloy 910, Alumide, Nylon, PLA (Polylactic Acid) , HIPS (High Impact Polystyrene), PVA (Polyvinyl Alcohol) or Rubberlike you can print interlocking parts, while in others like Gold, Silver, Bronze, or SLA Resinthis is not possible. The reason behind this is not the material itself, but the technology that is used for printing each of these materials.
Where 3D-Printing is an Additive Manufacturing process.
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