Saves are compatible between versions. You can even load 0.01 version save and it should work for 0.49 version.
If your saves "disappear", you either play from incognito mode or your browser cache gets cleared, or at least for me, once windows update cleared cache too.
Anyways you should never start a new game with newer version, it should always work. If you play incognito mode, you can always use "save to disk" just to be sure your saves are safe.
Interesting. Through both Chrome and Firefox, my saves do not carry over between versions. And I have
never had Sugarcube games using the "save to disk" function, then the "load from disk" actually load any saves. It acts like it does something, but the save list remains empty afterwards. In both Chrome/Firefox I have Adblock plus and Privacy Badger installed. Privacy Badger shows it is blocking a cookie for GoogleTagManager
Current run in firefox 0.38a:
Then without closing firefox, opening 0.39:
This has been my experience with every Sugarcube/Twine based game. Regardless of browser used. Maybe it's ABP/Privacy Badger? But the saves persist in a given version throughout reboots/etc. They just don't survive changing versions.
Honestly, I don't expect anybody to solve this, just assumed this was a weird known bug of Sugarcube. Alas, sounds like it's just my problem/PEBKAC.
*Edit: Weird example. In 0.38a I do a Save to Disk. Then delete a few saves. Then do a Load from Disk. The ones deleted remain deleted. Nothing is restored by doing the Load from Disk, indicating it failed to actually import (or possibly export) the data.