"Save" only saves the tasks (not the times) in order to provide a way to save a default configuration. Saving tasks with time tracked on them wasn't a use case for me yet, but I'll think about that.
There's a safety prompt when closing the tab and/or browser window while there's an active task. That doesn't kick in when rebooting though.
Overall, it'll probably be best to add an options menu at some point that controls things like global auto-save, auto-start for new tasks and maybe some other things that pop up over time.