I was wondering what race he is. Is he an Altmer or Aldmer? Or is there any real difference between the two?

Altmer. The division between Aldmer and Altmer is unclear, as the Altmer essentially seemed to change by staying the same while others split off from their culture during the Merethic Era. "Aldmer" is also used generally to refer to elves up to modern times. So finding a precise time when one ends and the other begins is all but impossible, unless there's some ESO info I don't know about. Regardless, the Altmer had certainly emerged as a distinct race by the time Galerion was born.