The truth is that Fiddler on stage is significantly different from Fiddler on screen. This is always the case when filmmakers adapt stories from other media. When a long novel like Up in the Air is adapted for the screen, many characters and plot elements must be eliminated; when a short memoir like An Education is adapted, many background details must be added.
Theatrical adaptations present the added complexity of acts. Most musicals, including Fiddler, have two acts, with Act One deliberately ending on a high because the creators want you to dash back to your seat after intermission. Sitting at home watching a DVD, you control the starts and stops, which has a huge impact on the flow.
But beyond these generic considerations, the one factor that most clearly differentiates “Fiddler on stage” from “Fiddler on screen” is historical context.