What is the difference between QED and quantum optics? Once you have QED, you have not only the quantized EM field, but also the interaction between matter and the EM field. So, you should have quantum optics as a consequence too. So, what is the difference between them? I am not an expert in neither of them, I should say.
Quantum optics is a field/branch of physics, with QED being a specific theory. Those working in quantum optics use a fully quantum theory (QED) when needed, but otherwise resort to using simpler tools such as the semiclassical approximation (classical field and quantum matter) which accurately describe a wide range of experiments (primarily involving the coherent interaction of a laser beam with matter).
Quantum optics is a branch of AMO physics that is concerned with understanding the physics of light and how it interacts with matter at a fundamental level.
The "quantum" part of "quantum optics" can be concerned with the quantum nature of light, in which case you would be talking about various (equivalent) descriptions of QED. However, many areas of quantum optics are concerned with how light interacts with matter. In such cases it is the quantization of the matter that makes it "quantum optics," regardless of whether one is using a quantum or classical field to describe the process.
When the field is treated as classical, but the matter is quantized, this is called a semiclassical description. Semiclassical quantum optics descriptions turn out to be incredibly powerful due to the fact that classical fields can be fully coherent, and thus you can control the quantum coherence properties of matter using only a classical EM field.
In practice the reason why semiclassical optics is such a major field in physics is due to the invention/existence of the laser, which is an extremely powerful, yet classical device, that allows you to do all sorts of quantum experiments (many of which are much more difficult or even impossible to do in non optics based setups).
Fully quantum descriptions (e.g. QED) are only necessary in a subset of experiments. Some typical examples of when this is necessary are situations involving (photon) counting statistics, fundamental/quantum limited noise, or experiments specifically looking at the physics of nonclassical light.