C# DirectX API Face-off: SlimDX vs SharpDX – Which should you choose?

A question I often see asked by beginning game programmers on the internet is:

I want to code my game in C#. Which DirectX wrapper API should I use?

As with most such things, all choices have their pros and cons. Here I will look at the two frameworks most currently in use at the time of writing: SlimDX and SharpDX.