Java applets would seem to have their best analogies to WPF Browser Applications in .NET, if not Silverlight 2.0. There's no perfect mirror between Java and .NET in this respect - some technolgoies seem to be more similar to each other in certain respects and others in different respects. Given that Java was developed largely for the purpose of applets, and the .NET Framework for desktop applications, there are naturally going to be fundamental difference. Although WPF browser applications are of course restricted to the Windows platform (unlike Silverlight), they perhaps resemble applets more greatly in the respect that they can utlise the entire .NET Framework, among other things. Also, as has been pointed out, Silverlight is more analogous to JavaFX.
In terms of servlets, the equivalent is effectively the whole of ASP.NET (moreover the Web Application side as opposed to websites), though that is being slightly vague. Perhaps more accurately, JavaServer Pages most resembles ASP.NET (either WebForms or MVC [Model-View-Controller] for that matter). In the case of the former (Java), content is compiled into Java servlets, whereas in the case of the latter (.NET), content is compiled into .NET assemblies. So maybe .NET assemblies of web applications are most akin to servelets - though to be honest I don't know enough about the Java side of things to make much of a conclusion.
Interestingly, the histories of .NET and Java began somewhat differently (admittedly they were both VM frameworks, and Java inspired .NET), nonetheless in many aspects they have converged over time so that nowadays you'll pretty much find an equivalent technology in either of the two frameworks, though often in one there has been significantly more development and/or success (Silverlight is one example in favour of Microsoft, whereas applets are perhaps in favour of Sun). Anyway, hopefully I've at least provided an overview of where the similarities and differences lie in the two technologies.