This Ableton tutorial walks through the three core ways to save a project in Live: Save Live Set, Save Live Set As, and Save a Copy. Each serves a distinct purpose, and mixing them up can cost you work or create a mess of files.
Save Live Set (Cmd+S on Mac, Ctrl+S on PC) captures everything in your session: clips, their positions, and all device settings. You can run it at any point, even during playback.
Save Live Set As creates a new version of the same set inside the same project folder, useful when you want to branch off without leaving your existing project structure. Save a Copy goes further, duplicating the entire set, and it's worth pointing to a new destination so Live generates a separate project folder for it rather than mixing files together.