C++ is a superset of C, so if you were to learn all of the C++ constructs, you would then, by definition, know C.
That said, there are several constructs in C++ that replace the C versions. Examples include dynamic memory allociation (new and delete instead of malloc and free), the handling of files (streams instead of file pointers), and the handling of strings (STL string and stream classes instead of character arrays).
Also, if you are using C++ to its fullest extents, you are using Object Oriented principals. C is, by its nature, a structured language. You can do structured C++, but to do OO C requires some really awkward abstraction.