I've been riding road bikes for years but really wasn't too aware that there appear to be two different tire sizes, for example a 23C and a 23. ( 700 x 23. And 700 x 23c )
They are indeed the same size of tire. For some reason, the "c" gets shuffled about, sometimes being put after the 700, sometimes being put after the tire width (in this case, 23mm), etc.
Peruse Sheldon Brown's Tire Cribsheet, and if you want lots of fun facts about tire sizing (including the difference between French standards and ISO standards), read Sheldon's article on Tire Sizing.