Is there a difference between being anxious to do something and being eager to do something?

Yes.  To be anxious is to be nervous, worried, or concened about something; to be eager is to look forward to something.  In addition, anxious is usually followed by about, whereas eager is usually followed by to.  The most common mistake here is to say something such as, "I am anxious to go on vacation."  If you are nervous, worried, or concerned about going on vacation, you might be anxious about going on vacation; however, if you are looking forward to going on vacation, as is probably the case, you want to say that you are eager to go on vacation.