Adapted, from my inbox:
These two words often are used to mean the same thing, but they do not.
Anxious: Full of mental distress because of apprehension of danger or misfortune [in effect, seeking to avoid] (We were anxious about the nearby gunshots.)
Eager: looking forward to something (Until you returned, I was anxious to see you.)
Thus, you cannot accurately write "They were anxiously awaiting his arrival" unless the persons arrival had toe potential of a negative experience. It is likely more accurate and appropriate to write, "They were eagerly awaiting his arrival."
Now go, and write better.