While this may be fairly obvious, the first difference between the movie and the comic book is the title itself. While the story in the Batman comics by Snyder and Capullo ran as "The Court of Owls," the title "Batman vs Robin" came from the first volume of the
"Batman and Robin" series, written by Grant Morrison and illustrated by Andy Clarke. As a part of Morrison's brilliant Batman saga, this story saw a reluctant Damian Wayne -- under the control of Deathstroke -- face off against Dick Grayson, who was under the Batman cowl at the time.
For the movie, the title pointed to the divide and confrontation between Bruce and Damian as the Talon tried to convince Damian to join his side, but it had nothing to do with either comic series it borrowed from. Damian wasn't controlled by Slade Wilson to fight his father -- that's something he did willingly in the film. The movie may have taken cues from the two series, but it ended up doing neither of them justice.