The Google, Yahoo!, Bing, and ESRI Mappings API allow you to connect with their map server backend. Their APIs also usually provide a client side interface (at least in the case of Google Maps).
The Google Maps API, for instance, is fairly powerful. You have the ability to add markers, plot routes, and use KML data (things you can also do in OpenLayers)—but the main drawback is that your mapping application relies totally on Google. The map client and map server are provided by a third party. This is not inherently a bad thing, and for many projects, Google Maps and the like are a good fit.
However, there are quite a few drawbacks.