In careful use, anxious implies anxiety, worry, or uneasiness: "The physician was anxious about the pulse rate of his patient." Eager means "keenly desirous," "wanting to": "Bob was eager to see his old friend." Eager is rarely used where anxious is meant, but anxious is often incorrectly substituted for eager: "The small boy was eager (not anxious) to go fishing." One is anxious about something of which he is fearful. He is eager concerning something looked forward to.
Dictionary of problem words and expressions. Harry Shaw. 1975.