NodeWebKit is a platform and offer file access. However NodeWebKit plugin only works on the NodeWebkit platform. If you are using say Chrome, Crosswalk, Ejecta, HTML.... everything but NodeWebkit platform then the plugin won't store anything.
So the question isn't which is stable(which is the same). It's what do you need? If your going to run on anything else than the NodeWebKit platform then you have to use WebStorage. If your only using NodeWebkit then do you need more advanced storage. I suggest reading up more on the plugins themselves to determine your needs.