First, my own question: if we combine the tags, do searches for the term "astrodynamics" suddenly return fewer results? If so, that would be bad, because so many textbooks have astrodynamics in the title. In that case, I would argue for always tagging with both together. However, I'm only guessing; does anyone know for sure?
If it doesn't screw up searches, I lean toward combining them. There is an argument for keeping them separate, but only if we are going to be clear, thorough, and vigorous in maintaining the separation.
There are many competing definitions of each term, ranging from identical to disjoint, with varying degrees of overlap in between. Where there are differences, the term "astrodynamics" is restricted to artificial bodies, while orbital mechanics includes also natural bodies. When there isn't a difference, it seems generally to be because that author also restricts orbital mechanics to artificial bodies.
If we believe astrodynamics means only artificial satellites, and orbital mechanics includes both those and natural satellites, then we could consider drawing and defending a line in the sand to keep one a proper subset of the other, but it would require vigilance. Alternately, if astrodynamics and orbital mechanics are both restricted to artificial bodies, what other tag should we use for planets, moons, comets, and asteroids?
This question on the main site regards the difference between astrodynamics and "space dynamics", and contains a couple useful links, including Vallado's definition and Wikipedia's (astrodynamics and orbital mechanics are equal).
Bate, Mueller, and White, Fundamentals of Astrodynamics (1971), don't seem to define astrodynamics explicitly, but they do say "our efforts in this text will be devoted to studying the motion of artificial satellites, ballistic missiles, or space probes" (section 1.3, page 13).
Pedro Ramon Escobal, Methods of Astrodynamics (1968) says "astrodynamics is the science of applying the techniques of celestial mechanics to the solution of space engineering problems."
Richard Battin, An Introduction to the Mathematics and Methods of Astrodynamics (1999) says "The term ''Astrodynamics'', attributed to the late Sam Herrick, came into common usage in the 1950s to categorize aspects of celestial mechanics relevant to a new breed --- the aerospace engineer."
Samuel Herrick, Astrodynamics (1971) has the most elaborate definition, and also the most amusing (bold and italics in the original):
Astrodynamics must be defined in terms of celestial mechanics on the one hand and space navigation on the other.
Celestial mechanics is concerned with the motions of objects in astronomical space, with the physical forces that govern these
motions, and with the mathematical assumptions, conditions, and
processes by which they are determined and predicted for observation
and correction. Like Gaul, celestial mechanics is divided into three
parts, and although a 'celestial mechanic' is not likely to be
concerned exclusively with one of these parts, his heart will
nevertheless be in one and he may be slightly less than clairvoyant in
Mathematical celestial mechanics... prefers general solutions over special solutions... on the other hand it is inclined to delimit a
problem in order to employ general analytical tools that may exclude
some real problems, or be awkward in handling them.
Physical celestial mechanics is concerned with explanations of observed phenomena, and with the evaluation of physical constants
associated with the formulated explanations, especially if these
constants are useful in other areas of astronomy, geophysics, or
physics in general...
Astrodynamics is a term that has come to designate the third field of celestial mechanics, which is concerned with the determination,
integration, and improvement of specific 'real-world' orbits. It is
fully aware of mathematical and physical celestial mechanics, and
frequently must solve its own problems in those fields. The
demonstration of the existence of a solution is often the end of
mathematical celestial mechanics and the beginning of astrodynamics...
Completing this definition, and then drawing parallels and contrasts with "space navigation", takes the next seven pages (the posted quote is about half a page).