In general (considering all languages and all compilers) a switch statement CAN SOMETIMES be more efficient than an if / else statement, because it is easy for a compiler to generate jump tables from switch statements. It is possible to do the same thing for if / else statements, given appropriate constraints, but that is much more difficult.
In the case of C#, this is also true, but for other reasons.
With a large number of strings, there is a significant performance advantage to using a switch statement, because the compiler will use a hash table to implement the jump.
With a small number of strings, the performance between the two is the same.
This is because in that case the C# compiler does not generate a jump table. Instead it generates MSIL that is equivalent to IF / ELSE blocks.
There is a "switch statement" MSIL instruction that when jitted will use a jump table to implement a switch statement. It only works with integer types, however (this question asks about strings).
For small numbers of strings, it's more efficient for the compiler to generate IF / ELSE blocks then it is to use a hash table.
When I originally noticed this, I made the assumption that because IF / ELSE blocks were used with a small number of strings, that the compiler did the same transformation for large numbers of strings.
This was WRONG. 'IMA' was kind enough to point this out to me (well...he wasn't kind about it, but he was right, and I was wrong, which is the important part)
I also made a bone headed assumption about the lack of a "switch" instruction in MSIL (I figured, if there was a switch primitive, why weren't they using it with a hash table, so there must not be a switch primitive.... ). This was both wrong, and incredibly stupid on my part. Again 'IMA' pointed this out to me.
I made the updates here because it's the highest rated post, and the accepted answer.
However,I've made it Community Wiki because I figure I don't deserve the REP for being wrong. If you get a chance, please up vote 'ima''s post.