There is no doubt that film can be very powerful. They can make you angry, make you cry, make you feel joy. I want to see what examples of interactive media you know of that can invoke emotion in a way that a film cannot.

The end of this I feel is an excellent example of how the interaction an not simply what you see and hear make a difference on what you feel. I don't want to give any spoilers, so I wont elaborate too much on it.
http://interactivehaiku.com/lifeisshort/

Do you know of any other examples where being the controlling agent changes your perception of the experience? Any moments in games that made a lasting impression where film would struggle to invoke the same emotion?

I felt like I better understood how gameplay can enhance your feeling of connection to a character from The Last of Us. You start the game playing as a little girl, the daughter of the main character. As a player you know that civilization is about to collapse (cause that's what the game is about), and you're just a powerless little girl walking through your house at night, looking for your dad. When you find him, you can tell he's worried, and you know you have to stay close to him because you're not capable of defending yourself. So later when you're playing as the dad, the game has already created a bond between you and the daughter. You feel like it's up to you to protect her now. I don't think that feeling would have been as strong if you were in control of the dad the whole time. I don't think it would have been as strong in a movie.

I think good voice acting and cut scenes are probably better at cashing in on the connection you feel with characters, but gameplay can help make those connections stronger, enhancing the emotions you feel from a scene.