C was the C++ predecessor. As it's name implies, alot of C remains in C++. Although not actually being more powerful than C, C++ allows the programmer to more easily manage and operate with Objects, using an OOP (Object Oriented Programming) concept.
C++ allows the programmer to create classes, which are somewhat similar to C structures. However, to a class can be assigned methods, functions associated to it, of various prototypes, which can access and operate within the class, somewhat like C functions often operate on a supplied handler pointer.
Although it is possible to implement anything which C++ could implement in C, C++ aids to standarize a way in which objects are created and managed, whereas the C programmer who implements the same system has alot of liberty on how to actually implement the internals, and style among programmers will vary alot on the design choices made.
In C, some will prefer the handler-type, where a main function initializes a handler, and that handler can be supplied to other functions of the library as an object to operate on/through. Others will even want to have that handler link all the related function pointers within it which then must be called using a convention closer to C++.
To finish this discussion, C++ applications are generally slower at runtime, and are much slower to compile than C programs. The low-level infrastructure for C++ binary execution is also larger. For these reasons C is always commonly used even if C++ has alot of popularity, and will probably continue to be used in projects where size and speed are primary concerns, and portable code still required (assembly would be unsuitable then).
