What differs is the locations in which the preprocessor searches for the file to be included.
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#include <filename>The preprocessor searches in an implementation-defined manner, normally in directories pre-designated by the compiler/IDE. This method is normally used to include header files for the C standard library and other header files associated with the target platform. -
#include "filename"The preprocessor also searches in an implementation-defined manner, but one that is normally used to include programmer-defined header files and typically includes same directory as the file containing the directive (unless an absolute path is given).
For GCC, a more complete description is available in the GCC documentation on search paths.