The deleted scene in question shows Ripley shortly after waking up from her stasis pod, as she asks Carter Burke (Paul Reiser), "Do you have any news about my daughter?" Burke sits her down and calmly explains that Ripley's daughter has been dead for two years, explaining she died at 66. Her decision to go into a stasis sleep in the first movie takes on an entirely new light — and we as an audience understand exactly what she's sacrificed fighting these aliens.
It's a complete gut-punch to Ripley, made even more heart-wrenching when she quietly says, "I promised her I'd be home for her birthday. Her 11th birthday," before breaking down and crying in earnest. It seems like a significant piece of character development for Ripley and casts the entire movie in a new light.
The scene makes Newt more than just a girl Ripley's trying to save — she's a representation of the daughter she lost, and saving her becomes an atonement of Ripley's guilt as much as a weight on her conscience. It's such an essential scene in retrospect, and it's very strange that it was left out of the theatrical release as it casts Ripley in an entirely different light.
