Runs in progress cannot be saved in either version.
Unlocks, stats, saved run variants and preferences should persist on the web version, in the form of a cookie stored on your machine. However, every web browser handles cookies differently. I have heard of people's cookies lasting upwards of a month, and others get deleted as soon as they close the tab.
The Windows version stores these files in C:\Users\UserName\AppData\LocalLow\SilverDubloons\Scrongly