LeafletJS is awesome. It's an important open source offering within the OpenStreetMap ecosystem. If you're looking for an "OpenStreetMap API" for embedding a map, then you're looking for LeafletJS! ...or another one called OpenLayers, or various other lesser used options. You have a choice! With OpenStreetMap you also have a choice of where you get your map "tiles" from, or you can render and host them yourself. Check out this website which explains a lot more about the various options for making the switch to OpenStreetMap:

http://switch2osm.org