longing is specifically 'looking FORWARD to' , wishing fervently for something, I feel it has a sense of yearning, something is missing... Eager is keen or impatient, have a strong desire, it doesn't have that feeling of something missing or lost. They are both used commonly enough, and really used in different situations. I am eager to start University again. I am eager to see my new touchscreen. But! I am longing for my next trip to Guilin, I'm longing for my next trip overseas.

I guess (be longing) is for missing someone or a place or an activity. For example : I'm longing for you= I miss you...OR.. I'm longing for coming back home = I miss my house. BUT (be eager) means has passion and intense desire to do something. Ex: I'm eager to work = I'm so excited to work. OR I'm eager to fulfill my dream = I want my dream to come true so badly! < I wish I'm not giving wrong information but this is how I learnt it.. :D if im mistaken please correct me