As Elizabeth Barrett Browning said: How do I love thee? Let me count the ways.
Please excuse the qualitative (vs. quantitative) aspect of this post.
Comparing these 2 languages (and their associated run-times) is very difficult. Comparisons can be at many levels and focus on many different aspects (such as GUI development mentioned in earlier posts). Preference between them is often personal and not just technical.
C# was originally based on Java (and the CLR on the JRE) but, IMHO, has, in general, gone beyond Java in its features, expressiveness and possibly utility. Being controlled by one company (vs. a committee), C# can move forward faster than Java can. The differences ebb and flow across releases with Java often playing catch up (such as the recent addition of lambdas to Java which C# has had for a long time). Neither language is a super-set of the other in all aspects as both have features (and foibles) the other lacks.
A detailed side-by-side comparison would likely take several 100s of pages. But my net is that for most modern business related programming tasks they are similar in power and utility. The most critical difference is probably in portability. Java runs on nearly all popular platforms, which C# runs mostly only on Windows-based platforms (ignoring Mono, which has not been widely successful). Java, because of its portability, arguably has a larger developer community and thus more third party library and framework support.
If you feel the need to select between them, your best criteria is your platform of interest. If all your work will run only on Windows systems, IMHO, C#/CLR, with its richer language and its ability to directly interact with Windows' native APIs, is a clear winner. If you need cross system portability then Java/JRE is a clear winner.
PS. If you need more portable jobs skills, then IMHO Java is also a winner.