There should be very little, if any, performance difference, to the point that if there is a difference, it will be very nearly immeasurably small - assuming you're using a C and C++ compiler from the same vendor or collection. Using a C compiler from one vendor and a C++ compiler from another will likely show larger differences, but only due to the fact that different vendors implement different optimization strategies.
There are a small number of potential optimization opportunities that C++ calls for that C compilers may not natively support - but again, unless they're from different vendors, the difference will be meaningless, and many compilers implement similar optimizations in both C++ and C compiler frontends.
One exception to this is Microsoft's compiler - Microsoft never made a C compiler, to my knowledge.
Note: I am assuming that the code does not use C++-specific features like templates or classes.