Why are there buttons and fields in the web editor for a lot of information that is never used on the map?

Because OpenStreetMap is much, much more than the map on OpenStreetMap.org! In fact, one could argue that the map is merely a tool to motivate and help OpenStreetMap (OSM) mappers to collect geographical information. While making a visual street map (similar to for example Google Maps) is one of the things the OpenStreetMap community does, I'd say the primary purpose is to create a free and open repository of geographical information so that others can make street maps and all other imaginable and unimaginable maps and geo-services.

So, while you cannot see all the details you enter on the OpenStreetMap map itself, you'll find speed cameras and fuel stations with opening hours and contact information in mobile apps such as OsmAnd and Mapdroyd, information about wheelchair access on wheelmap.org, hiking information on Waymarked Trails (including website information) and turkompisen.no (only Norway), cycle parking with dedicated cycle apps (example only for Britain), and so on and so forth. You can also make your own map where you use the information important to you. On the wiki we have a list of services based on OSM, and software that can use OpenStreetMap data.

Oh, and you can see all the information you have provided on the OpenStreetMap map! Just zoom in so you can see the details of an area, click on the triangle right of "Edit", and choose "Browse Map Data".