With Astral Projection you get into the Astral Plane direct from your physical body. When you have a Lucid Dream, you have simply ‘missed’ the initial projection. They are just two different ways of entering the Astral Realm. In either case you are still conscious in the Astral, however, as there are degrees of lucidity in dreaming, you may or may not be fully in control of your experience. But this is highly unlikely in an Astral Projection. There is still a difference, whether more slight or obvious.
The difference arises mostly because of the framework you are entering from. If you project from the waking state, you may encounter some difficulties due to restrictive ideas carried with you that don’t apply to the Astral, and require some practice to adjust to the new environment. On the other hand, in a Lucid Dream, having accustomed to the expansiveness of the dream state, you can sometimes have the opposite problem, finding it hard to retain a conscious focus, and thus ‘losing lucidity’. With practice of course, both of these will be overcome.
However, knowing this distinction can be helpful when choosing our experience. In one sense, when we are Lucid Dreaming we are within the ‘creative state’, and it is easier to consciously create what we desire. And in Astral Projection it is probably more productive to explore the other realities that already exist, relatively unhampered by specific desires for self-fulfillment.
There is some contention between defining the difference between Lucid Dreaming and Projection. It is not a matter of clarity or vividness as some people suggest, though typically Lucid Dreams will seem more substantial and realistic at least to begin with. But either experience can be poorly or highly developed. Having already distinguished Etheric Projection and Astral Projection, there remains a factor that does provide a feasible distinction.
This determining factor is memory. While there are no hard and fast rules, typically, in an Astral Projection you remember who you are, and there is a continuity of memory from waking through to projection. But in a Lucid Dream, this is not always the case. You can be consciously aware without fully remembering your identity, and there is often no continuity from the waking state, nor memory of what preceded the experience of becoming lucid.