Warning: The question you've asked is really pretty complex -- probably much more so than you realize. As a result, this is a really long answer.
From a purely theoretical viewpoint, there's probably a simple answer to this: there's (probably) nothing about C# that truly prevents it from being as fast as C++. Despite the theory, however, there are some practical reasons that it is slower at some things under some circumstances.
I'll consider three basic areas of differences: language features, virtual machine execution, and garbage collection. The latter two often go together, but can be independent, so I'll look at them separately.
Language Features
C++ places a great deal of emphasis on templates, and features in the template system that are largely intended to allow as much as possible to be done at compile time, so from the viewpoint of the program, they're "static." Template meta-programming allows completely arbitrary computations to be carried out at compile time (I.e., the template system is Turing complete). As such, essentially anything that doesn't depend on input from the user can be computed at compile time, so at runtime it's simply a constant. Input to this can, however, include things like type information, so a great deal of what you'd do via reflection at runtime in C# is normally done at compile time via template metaprogramming in C++. There is definitely a trade-off between runtime speed and versatility though -- what templates can do, they do statically, but they simply can't do everything reflection can.
The differences in language features mean that almost any attempt at comparing the two languages simply by transliterating some C# into C++ (or vice versa) is likely to produce results somewhere between meaningless and misleading (and the same would be true for most other pairs of languages as well). The simple fact is that for anything larger than a couple lines of code or so, almost nobody is at all likely to use the languages the same way (or close enough to the same way) that such a comparison tells you anything about how those languages work in real life.
Virtual Machine
Like almost any reasonably modern VM, Microsoft's for .NET can and will do JIT (aka "dynamic") compilation. This represents a number of trade-offs though.
Primarily, optimizing code (like most other optimization problems) is largely an NP-complete problem. For anything but a truly trivial/toy program, you're pretty nearly guaranteed you won't truly "optimize" the result (i.e., you won't find the true optimum) -- the optimizer will simply make the code better than it was previously. Quite a few optimizations that are well known, however, take a substantial amount of time (and, often, memory) to execute. With a JIT compiler, the user is waiting while the compiler runs. Most of the more expensive optimization techniques are ruled out. Static compilation has two advantages: first of all, if it's slow (e.g., building a large system) it's typically carried out on a server, and nobody spends time waiting for it. Second, an executable can be generated once, and used many times by many people. The first minimizes the cost of optimization; the second amortizes the much smaller cost over a much larger number of executions.
As mentioned in the original question (and many other web sites) JIT compilation does have the possibility of greater awareness of the target environment, which should (at least theoretically) offset this advantage. There's no question that this factor can offset at least part of the disadvantage of static compilation. For a few rather specific types of code and target environments, it can even outweigh the advantages of static compilation, sometimes fairly dramatically. At least in my testing and experience, however, this is fairly unusual. Target dependent optimizations mostly seem to either make fairly small differences, or can only be applied (automatically, anyway) to fairly specific types of problems. Obvious times this would happen would be if you were running a relatively old program on a modern machine. An old program written in C++ would probably have been compiled to 32-bit code, and would continue to use 32-bit code even on a modern 64-bit processor. A program written in C# would have been compiled to byte code, which the VM would then compile to 64-bit machine code. If this program derived a substantial benefit from running as 64-bit code, that could give a substantial advantage. For a short time when 64-bit processors were fairly new, this happened a fair amount. Recent code that's likely to benefit from a 64-bit processor will usually be available compiled statically into 64-bit code though.
Using a VM also has a possibility of improving cache usage. Instructions for a VM are often more compact than native machine instructions. More of them can fit into a given amount of cache memory, so you stand a better chance of any given code being in cache when needed. This can help keep interpreted execution of VM code more competitive (in terms of speed) than most people would initially expect -- you can execute a lot of instructions on a modern CPU in the time taken by one cache miss.
It's also worth mentioning that this factor isn't necessarily different between the two at all. There's nothing preventing (for example) a C++ compiler from producing output intended to run on a virtual machine (with or without JIT). In fact, Microsoft's C++/CLI is nearly that -- an (almost) conforming C++ compiler (albeit, with a lot of extensions) that produces output intended to run on a virtual machine.
The reverse is also true: Microsoft now has .NET Native, which compiles C# (or VB.NET) code to a native executable. This gives performance that's generally much more like C++, but retains the features of C#/VB (e.g., C# compiled to native code still supports reflection). If you have performance intensive C# code, this may be helpful.
Garbage Collection
From what I've seen, I'd say garbage collection is the poorest-understood of these three factors. Just for an obvious example, the question here mentions: "GC doesn't add a lot of overhead either, unless you create and destroy thousands of objects [...]". In reality, if you create and destroy thousands of objects, the overhead from garbage collection will generally be fairly low. .NET uses a generational scavenger, which is a variety of copying collector. The garbage collector works by starting from "places" (e.g., registers and execution stack) that pointers/references are known to be accessible. It then "chases" those pointers to objects that have been allocated on the heap. It examines those objects for further pointers/references, until it has followed all of them to the ends of any chains, and found all the objects that are (at least potentially) accessible. In the next step, it takes all of the objects that are (or at least might be) in use, and compacts the heap by copying all of them into a contiguous chunk at one end of the memory being managed in the heap. The rest of the memory is then free (modulo finalizers having to be run, but at least in well-written code, they're rare enough that I'll ignore them for the moment).
What this means is that if you create and destroy lots of objects, garbage collection adds very little overhead. The time taken by a garbage collection cycle depends almost entirely on the number of objects that have been created but not destroyed. The primary consequence of creating and destroying objects in a hurry is simply that the GC has to run more often, but each cycle will still be fast. If you create objects and don't destroy them, the GC will run more often and each cycle will be substantially slower as it spends more time chasing pointers to potentially-live objects, and it spends more time copying objects that are still in use.
To combat this, generational scavenging works on the assumption that objects that have remained "alive" for quite a while are likely to continue remaining alive for quite a while longer. Based on this, it has a system where objects that survive some number of garbage collection cycles get "tenured", and the garbage collector starts to simply assume they're still in use, so instead of copying them at every cycle, it simply leaves them alone. This is a valid assumption often enough that generational scavenging typically has considerably lower overhead than most other forms of GC.
"Manual" memory management is often just as poorly understood. Just for one example, many attempts at comparison assume that all manual memory management follows one specific model as well (e.g., best-fit allocation). This is often little (if any) closer to reality than many peoples' beliefs about garbage collection (e.g., the widespread assumption that it's normally done using reference counting).
Given the variety of strategies for both garbage collection and manual memory management, it's quite difficult to compare the two in terms of overall speed. Attempting to compare the speed of allocating and/or freeing memory (by itself) is pretty nearly guaranteed to produce results that are meaningless at best, and outright misleading at worst.
Bonus Topic: Benchmarks
Since quite a few blogs, web sites, magazine articles, etc., claim to provide "objective" evidence in one direction or another, I'll put in my two-cents worth on that subject as well.
Most of these benchmarks are a bit like teenagers deciding to race their cars, and whoever wins gets to keep both cars. The web sites differ in one crucial way though: they guy who's publishing the benchmark gets to drive both cars. By some strange chance, his car always wins, and everybody else has to settle for "trust me, I was really driving your car as fast as it would go."
It's easy to write a poor benchmark that produces results that mean next to nothing. Almost anybody with anywhere close to the skill necessary to design a benchmark that produces anything meaningful, also has the skill to produce one that will give the results he's decided he wants. In fact it's probably easier to write code to produce a specific result than code that will really produce meaningful results.
As my friend James Kanze put it, "never trust a benchmark you didn't falsify yourself."
Conclusion
There is no simple answer. I'm reasonably certain that I could flip a coin to choose the winner, then pick a number between (say) 1 and 20 for the percentage it would win by, and write some code that would look like a reasonable and fair benchmark, and produced that foregone conclusion (at least on some target processor--a different processor might change the percentage a bit).
As others have pointed out, for most code, speed is almost irrelevant. The corollary to that (which is much more often ignored) is that in the little code where speed does matter, it usually matters a lot. At least in my experience, for the code where it really does matter, C++ is almost always the winner. There are definitely factors that favor C#, but in practice they seem to be outweighed by factors that favor C++. You can certainly find benchmarks that will indicate the outcome of your choice, but when you write real code, you can almost always make it faster in C++ than in C#. It might (or might not) take more skill and/or effort to write, but it's virtually always possible.
