After that scene ends, when Cinderella realizes it's midnight and tries to leave, she tells the Prince she's rushing because she hasn't met the Prince yet. She actually had no idea she was dancing with said prince and fell in love with him, and that's one of the reasons why Prince Charming fell for her. She spent all that time with him because of him, not that he was the prince.
In a deleted scene, Prince Charming is seen apparently hunting a deer, but it turns out they were friends and playing a game together.
All the animals come to Cinderella's aid by getting the key to unlock her door and get her to the Grand Duke before it's too late.
When Cinderella makes it downstairs just in time, it's very sweet to see the Duke kindly greeting her and giving her a chance at the slipper, even though Lady Tremaine and the stepsisters insist she's "just Cinderella" and couldn't possibly be the right girl. After a movie of Cinderella being ignored and treated horribly, it's very nice to see.
In another deleted scene, Prince Charming is reintroduced to Cinderella after she fits the slipper. While surprised that she's a servant girl, he accepts her immediately as his princess.
The Triumphant Reprise of "A Dream is a Wish Your Heart Makes" after Cinderella marries Prince Charming.