when you use plex.tv hosted web app all the data content (posters,summaries, when you stream things) is all over local. it is not going out to the internet and back in. just the interface the web app and code of the page is. there are some reasons you may need to use it ( chrome cast) but also it will at rare times be use to test a feature before deploying to the local app
you can use the local web app by going to/bookmarking serverip:32400/web where serverip is the local ip address of your server computer. ( if you are actually using your server computer you can go to 127.0.0.1:32400/web or localhost:32400/web )