Within the context of Aeronautics, there are seemingly different meanings corresponding to Space Dynamics and Astrodynamics. What is the difference between them in an engineering perspective, such as spacecraft dynamics and control?
Vallado begins  by borrowing Griffin and French's definition of astrodynamics as "...the study of the motion of man-made objects in space, subject to both natural and artificially induced forces." Vallado uses this definition because other related subjects - e.g., orbital dynamics, attitude dynamics - don't solely encapsulate what astrodynamics is. In the context of your question, one could claim that space dynamics is covered under the umbrella term of astrodynamics.
Most literature equates space dynamics with the kinematics of a problem, while astrodynamics covers both the kinetics and the kinematics. From an engineering perspective, think of this as space dynamics dealing with the right-hand-side of $F = ma$, while astrodynamics develops the left-hand-side and (attempts to) finds a solution to the overall system.
This question will likely result in a range of answers because terminology ranges from country to county, institution to institution and person to person. However the etymology of the two phrases could add some knowledge:
Astrodynamics; Astro - star, dynamic - study of bodies in motion Space dynamics; space - area outside the atmosphere of planetary bodies
However the definition of astrodynamics is frequently used as a coverall for the dynamics of orbiting bodies (at least my my experience). So I would suggest the following definition:
Astrodynamics: The study of interactions between bodies at motion in space.
Space dynamics: The study of bodies at motion in space.
The above would suggest astrodynamics is a subset of space dynamics.
I first want to comment a bit on the descriptive context given for the question. (I don’t have enough points to post a comment to the description.)
“Within the context of Aeronautics, …” The whole of human aerospace activity (engineering, operations, …) can basically be divided into two domains: aeronautics and astronautics. The subject matter of the question clearly falls into the latter rather than the former.
I’m going to take the answer by @jah138 as the correct one. I’ll add a bit of perspective from having taken courses in astrodynamics in the 1970s, followed by some developments after that time.
There is certainly an overlap between astronomical methods and astronautics when it comes to determining the positions of gravitational bodies in space. A set of “ephemerides”, i.e., trajectories over time, is kept for such bodies in the solar system. Astrodynamics encompasses computing trajectories, further refining them based on observational data, and planning/modifying the trajectories of essentially zero-mass (when compared to a planetary body) human-created objects to take advantage of the gravitational field defined by them.
Some interesting additions have shown up since the 70s. There are now interesting small secondary effects, such as the pressure of particles of solar wind, the continuing photonic pressure from the Sun itself, and heating of the Sun-facing sides of asteroids. From an engineering perspective, a light sail is an attempt to increase the relative impact of the photonic component on the spacecraft, thus adding a non-trivial term in trajectory computation. There is also a proposal to use the mass of a spacecraft to affect the trajectory of an asteroid.
In interplanetary space, this leaves the heavily dominating force as gravity from planetary bodies and the Sun. For satellites in low Earth orbit, particularly, as low as the International Space Station, there is a component of atmospheric drag and the influence of the Moon's gravitational field. Because spacecraft trajectory computation integrates accelerations over long periods of time, the minor components add up, and methods of improving precision become important.
I have not worked specifically on “space dynamics” problems, but clearly there is a whole area of spacecraft motion within an inertial frame of reference (i.e., a frame experiencing no accelerations), and how the spacecraft reacts in that frame. Within it, you might compute or measure the impact of reaction wheels, articulation of booms, or spinning of a spacecraft on an axis. All these fall into the kinematics of the spacecraft. To get to a mid-course correction of a spacecraft, you will need to compute a way to orient the spacecraft to impart a desired delta-V in a certain time window. This is clearly a dynamics problem, but not normally what I include in astrodynamics. Space dynamics seems like a perfectly reasonable descriptive term for this.
Compared with conventional astrodynamics problems, these feel like short-term effects. But their long-term impact is found in observational data over extended periods of time that go into astrodynamics computation problems.
Astrodynamics includes all forms of movement in a near zero gravity environment. Space dynamics is in part of the grater discipline of Astrodynamics. The differences are just a way of the words and how they are explained. Splitting Space Dynamics and Astrodynamics is just to create differences which do not exist. That is why the definition is "Astrodynamics The Movement of Mass in a Near Zero Gravity Environment" not man made and natural mass, but both man made and natural. Therefore this discussion dot a valid discussion. Henry Hislop Astrodynamicist Cambridge United Kingdom