The way I most often hear the words used (although not the only definitions):
montage: in film, a series of shots edited together to tell a story or create a mood.
collage: in visual art, a work made by cutting up different materials into pieces and putting them together to make a new picture.
More detailed definitions of montage and collage from Merriam Webster. Wikipedia has an entry for each word, too.
Just for fun, here is a parody of a training montage, where a movie shows characters becoming experts in just a few minutes of screen time.
Student or Learner