The European version has more scenes of Leon and Matilda going out on their hit jobs, and it's more darkly funny, as well as developing their relationship more closely beyond the scenes in his apartment or the hotel room. The scene where they go on their first job together and he's coaching her as the target looks confused as hell was funny. I could see why Portman's parents had requested some scenes to be cut, as there are moments where Matilda is flirting with Leon and trying to kiss him and he backs off. There is a good scene in a restaurant where they go out to celebrate a successful hit and she's getting drunk on champagne and alternating between acting like a giggly little kid and trying to act like what she thinks a grown, mature lover would be like, which comes off as uncomfortable as you'd think.
It is weird to write about it without it sounding gross, but in context, it made sense. I also really liked a scene that goes into Leon's backstory with an old girlfriend of his from his youth, explaining part of his reason why he is resistant to love and keeps to himself a lot with few friends. It's a great movie, but these extra scenes made it even better for me.