The biggest change with the Twine editor is that it's now web-based. This means that it's usable on Linux computers out-of-the-box (it used to require manually installing some extra software dependencies) as well as tablets. Instead of opening files saved to your computer, Twine stores your work in your web browser – see here for more details.
This doesn't mean you can't save your story to a file, of course. You can publish a story to a file at any time using the story menu, or quickly save all of your work to a single archive file for backup purposes using the Archive button in the story list.
Some smaller changes to the editor:
Start, you can make any passage your story's starting point. Use the rocket-ship icon in the passage menu to mark it as such.
StoryTitlepassage, you change your story's name using the story menu.
Twine 2 doesn't use .tws story source files like Twine 1 did. Instead, Twine 2 embeds all of the information needed to import a story into the editor into the published .html file, That means you can open your story file in a web browser to read it, or import it right back into the editor.
There are also Twine archive files, which are created by using the Archive button in the story list. All they are are the HTML for all of your stories embedded into a single file.
Instead of Jonah, SugarCane, and the Responsive story formats, there are three new built-in formats.
If you're comfortable with the Twine 1 scripting syntax, using SugarCube will offer the smoothest transition for you.
Not right now, unfortunately. This may change in the future.