Vallado begins [1] 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.
1: Vallado, David A. Fundamentals of Astrodynamics and Applications, 3rd Ed. 2007
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