In Microsoft's words, the .Net Framework consists of the following parts:
* Common Language Runtime or CLR, which provides an abstraction layer over the operating system
* Base Class Libraries, which are pre-built code for common low-level programming tasks
* Development frameworks and technologies, which are reusable, customizable solutions for larger programming tasks
Mono is a software platform designed to allow developers to easily create cross platform applications. Sponsored by Novell (http://www.novell.com/), Mono is an open source implementation of Microsoft's .NET Framework based on the ECMA standards for C# and the Common Language Runtime.
It provides decent support for the CLR but it is lacking the breadth of "Base Class Libraries" and "Development frameworks & technologies". That is especially true if one considers the enormous ecosystem of Java. The latter has achieved a level of maturity and is supported by a plethora of publications (Wikis, User Guides, Books, Articles, etc.). Lastly, there has been a number of new languages (e.g. Groovy, Scala) and frameworks (e.g. Grails, Spring Roo, GWT) that can make your team enormously productive in a wide area of functional domains.
The only caveat would be, of course, if you must integrate Microsoft products (e.g. Microsoft Office) in your application (whether within the browser or a stand-alone app). In that case the .NET approach would give you a significant advantage but it would not be easy to carry over these advantages to other platforms (e.g. Linux).
The Java ecosystem is enormous and contains high-quality libraries for pretty much anything that you can think of. In fact, before you start writing your app., you should conduct a search across the Open Source projects that have been written in Java. It is very likely that what you are trying to build is very similar to something that already exists. So, depending on the license (look for LGPL, BSD, MIT, etc.), you might find something to get you started before you even break a sweat.
In summary, if your primary goal is to achieve multi-platform support then Java is your best bet.