I work for a tech company that does more prototyping than product shipment. I just got asked what's the difference between C# and F#, why did MS create F# and what scenarios would it be better than C#.
I've been using the language for a while now and I love it so I could easily go on about the great features of F# however I lack the experience in C# to say why we should use one over the other.
What's the benefits of using C# vs F# or F# vs C#?
General benefits of functional programming over imperative languages:
You can formulate many problems much easier, closer to their definition and more concise in a functional programming language like F# and your code is less error-prone (immutability, more powerful type system, intuitive recurive algorithms). You can code what you mean instead of what the computer wants you to say ;-) You will find many discussions like this when you google it or even search for it at SO.
Special F#-advantages:
Asynchronous programming is extremely easy and intuitive with async {}
-expressions - Even with ParallelFX, the corresponding C#-code is much bigger
Very easy integration of compiler compilers and domain-specific languages
Extending the language as you need it: LOP
More flexible syntax
Often shorter and more elegant solutions
Take a look at this document
The advantages of C# are that it's often more accurate to "imperative"-applications (User-interface, imperative algorithms) than a functional programming language, that the .NET-Framework it uses is designed imperatively and that it's more widespread.
Furthermore you can have F# and C# together in one solution, so you can combine the benefits of both languages and use them where they're needed.