They are slightly different. Let me add some context:
The plane was delayed and the hotel had given our rooms away but I didn't care, I was just happy to be there. (meaning I only needed the ONE thing to make me happy and that thing was "being there". The other things good or bad didn't matter. At the movie awards shows in the USA, the actors who don't win the award say "I was just happy to be nominated" meaning being nominated was enough to make me happy. I didn't need to win to be happy.
I was really happy to be there is simply an emphasis on the level of happiness. Same as I was very happy to be there. I was extremely happy to be there.