This is what I think (only an opinion):
Both are strong technologies.
That said, I think Java EE is more complicated than C#/ASP. When you develop a project in, let's say, Java Server Faces, you have nightmares trying to figure out which libraries and Frameworks work with which App Servers. And you end up having lots of strange problems and issues not directly related with what you're trying to accomplish: BUILDING A WEB SITE.
On the other hand, C# and ASP.NET, being from only one vendor, and working only on one platform (Windows with IIS {yes, I know there is MONO, but for practical purposes you could assume that it only works with Windows and IIS}) is a more predictable tool. You don't get many weird errors related to incompatibilities among AppServer+Libraries+Frameworks.
Java's strong side is Open Source tools availability. There are a lot of frameworks you could choose from, and is available on several platforms. And you can even run a server without spending a dime on software base (JBOSS Community+MySQL) while that's not true on .NET side.