Or is it now the other way around?
From what I've heard there are some areas in which C# proves to be faster than C++, but I've never had the guts to test it by myself.
Thought any of you could explain these differences in detail or point me to the right place for information on this.
There is no strict reason why a bytecode based language like C# or Java that has a JIT cannot be as fast as C++ code. However C++ code used to be significantly faster for a long time, and also today still is in many cases. This is mainly due to the more advanced JIT optimizations being complicated to implement, and the really cool ones are only arriving just now.
So C++ is faster, in many cases. But this is only part of the answer. The cases where C++ is actually faster, are highly optimized programs, where expert programmers thoroughly optimized the hell out of the code. This is not only very time consuming (and thus expensive), but also commonly leads to errors due to over-optimizations.
On the other hand, code in interpreted languages gets faster in later versions of the runtime (.NET CLR or Java VM), without you doing anything. And there are a lot of useful optimizations JIT compilers can do that are simply impossible in languages with pointers. Also, some argue that garbage collection should generally be as fast or faster as manual memory management, and in many cases it is. You can generally implement and achieve all of this in C++ or C, but it's going to be much more complicated and error prone.
As Donald Knuth said, "premature optimization is the root of all evil". If you really know for sure that your application will mostly consist of very performance critical arithmetic, and that it will be the bottleneck, and it's certainly going to be faster in C++, and you're sure that C++ won't conflict with your other requirements, go for C++. In any other case, concentrate on first implementing your application correctly in whatever language suits you best, then find performance bottlenecks if it runs too slow, and then think about how to optimize the code. In the worst case, you might need to call out to C code through a foreign function interface, so you'll still have the ability to write critical parts in lower level language.
Keep in mind that it's relatively easy to optimize a correct program, but much harder to correct an optimized program.
Giving actual percentages of speed advantages is impossible, it largely depends on your code. In many cases, the programming language implementation isn't even the bottleneck. Take the benchmarks at http://benchmarksgame.alioth.debian.org/ with a great deal of scepticism, as these largely test arithmetic code, which is most likely not similar to your code at all.
C# may not be faster, but it makes YOU/ME faster. That's the most important measure for what I do. :)