Yahoo, for example, will show you one of a possible 38,000 home pages, depending on who Yahoo believes you to be. Yahoo Mail (or any other mail, for that matter), is an utterly dependent service: it will only show youAyour mail (we hope). Facebook, of course, creates an entirely different experience for you than it does for me, because what Facebook shows us depends on who Facebook thinks we are. And search, in general, is a dependent service - what you see as results depends both on what you input as a query, as well as who the search service believes you are (personalized search).
And while I believe this idea of a dependent service being defined as "one that changes depending on its profile of you" is important, this isn’t the only feature that distinguishes Independent sites from Dependent ones.
Another way to understand the distinction is that Dependent sites tend to be ones we, well, depend on for some basic service in our lives. You might depend on Yahoo or Google for mail. We depend on Facebook for our social graph, and Twitter for our "interest graph." Of course we depend on Google (or Bing) for search. And I’m starting to depend on StumbleUpon to surface sites I might like.
In fact, most of us "depend" on Dependent-web services to discoverAindependent sites - a fact we may as well call "the interdependence of the independent and dependent web."