OSM doesn't have a server per se (other than the OSM API server). It's a dataset for the most part. Headway lets you bring up a server that will render maps tiles, allows you to search for places and get routes between places. It's trying to be a user-friendly UI for various pieces of the OSM ecosystem. There are a lot of amazing pieces of software out there that work with OSM data, but there isn't an easy way to bring them up, and there's no unified frontend for all of them other than Headway (that I know of).
Edit: To the comments telling me I'm wrong about this, remember that Headway has a geocoder and routing system, along with a transit trip planner (currently disabled on maps.earth though). These are not things you could get with off-the-shelf solutions that include a GUI until now. I also want to self-host my mapping software for privacy reasons, and Headway makes that easy too. I probably need to be better about explaining what all Headway does that's different than just bringing up an OSM tileserver though. Thank you all for the feedback :)