Flickr's API is a classic procedural library. There are calls for almost any action you could possibly do with all things Flickr : photos, account, tags, sets, etc. You can upload photos, access photo streams, search by tags ... pretty much anything you can do on Flickr's site. Each call is a different HTTP request. You can send the package of your request in a variety of formats: REST, XML-RPC, and SOAP. The REST and SOAP follow the usual patterns for these formats. The REST requests are "pure" HTTP requests. Flickr's XML-RPC is very true to the RPC mode. It uses simple XML to specify a procedure to call along with it's parameters.
The differences in these web APIs reflect the differences in these web services. Flickr is more of a pure data service. You can modify data (photos) and query that data (search for photos, etc.) Flickr wants to be your photo database. As such, they give you a number of ways to access your data. What you do with the data is up to you. Show your pics on a website, provide tools for uploading, print the pics, make aprons, whatever...
GMaps wants to draw maps for you. They don't want to give you mapping data for you to do whatever for it. They want to draw. They'll take your data and draw maps with it. They always draw the map, so they give your users a familiar interface that also integrates into Google. All roads lead back to Google.
Thus the APIs to these services are very appropriate for the services. I don't know for sure, but I'd wager they also reflect the programming design behind the services. It seems pretty obvious that Flickr is written in PHP and that Google Maps is written in Java (maybe the last one is less obvious, but I do have a little inside information on that one.) So it's not really surprising that a PHP site would have a very flexible, procedural API, and that a Java site would have more closed, object-oriented API.