Anxious and eager both mean ‘looking forward to something.’ But there is a difference in tone. Eager suggests a positive outlook and an enthusiasm about something. Anxious has a slightly negative connotation, implying worry about something.
So, you would be eager to get started with your vacation, but would be anxious to get all your work done before leaving for a vacation.
Here are two more examples:
Lily is very excited today. She is eager to get home and play with her new pet, a puppy. I’m anxious to get the puppy cleaned and fed before Lily arrives from school.