The differences between script writing and screenwriting all boils down to the fact that script writing only involves what is being said by and the description of the characters in the movie or TV show. It doesn't describe the setting and it doesn't provide the mood of what is being seen on the screen. Screenwriting builds upon the script to give a more vivid picture of what's happening on screen, serving the development and construction of the larger narrative.
Screenwriting deals more with how the story is going to be perceived by describing where the characters are, what they are doing, what they are saying and everything else that's happening around them. It may not even include character dialogues to accomplish this.
Script writing provides a narrative like a movie or a TV show with a blue print of the story. The script is what gives the characters what to say, but it doesn't involve giving them the proper mood or setting. The script usually goes from the script writing process to the screenwriting process, but sometimes, it can go the other way around. An idea for a movie or TV show may come first before the characters get their respective voices.