std::exception is the class whose only purpose is to serve as the base class in the exception hierarchy. It has no other uses. In other words, conceptually it is an abstract class (even though it is not defined as abstract class in C++ meaning of the term).
std::runtime_error is a more specialized class, descending from std::exception, intended to be thrown in case of various runtime errors. It has a dual purpose. It can be thrown by itself, or it can serve as a base class to various even more specialized types of runtime error exceptions, such as std::range_error, std::overflow_error etc. You can define your own exception classes descending from std::runtime_error, as well as you can define your own exception classes descending from std::exception.
Just like std::runtime_error, standard library contains std::logic_error, also descending from std::exception.
The point of having this hierarchy is to give user the opportunity to use the full power of C++ exception handling mechanism. Since 'catch' clause can catch polymorphic exceptions, the user can write 'catch' clauses that can catch exception types from a specific subtree of the exception hierarchy. For example, catch (std::runtime_error& e) will catch all exceptions from std::runtime_error subtree, letting all others to pass through (and fly further up the call stack).
P.S. Designing a useful exception class hierarchy (that would let you catch only the exception types you are interested in at each point of your code) is a non-trivial task. What you see in standard C++ library is one possible approach, offered to you by the authors of the language. As you see, they decided to split all exception types into "runtime errors" and "logic errors" and let you proceed from there with your own exception types. There are, of course, alternative ways to structure that hierarchy, which might be more appropriate in your design.
Update: Portability Linux vs Windows
As Loki Astari and unixman83 noted in their answer and comments below, the constructor of the exception class does not take any arguments according to C++ standard. Microsoft C++ has a constructor taking arguments in the exception class, but this is not standard. The runtime_error class has a constructor taking arguments (char*) on both platforms, Windows and Linux. To be portable, better use runtime_error.
(And remember, just because a specification of your project says your code does not have to run on Linux, it does not mean it does never have to run on Linux.)
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