What's the difference between the Riemann problem and the shock tube?

The Riemann problem is a method and the shock tube is an example of the method.

I guess you have confusions about the concept of the Riemann problem.

Riemann problem is a problem containing one single wave. That is,

u(x,t=0)=uL for x <x0

uR for x >x0 Like above, the problem is spatially divided by two region but constant. If it is not constant(i.e. uL(x),uR(x)), it is called the Generalized Riemann problem.

The shock tube problem is a very famous example used to simulate such a riemann problem.

The reason you confused may be the difference between the exact solution and the numerical solution of the Riemann problem. There are a number of books available on this issue. For example, computational gasdynamics written by Culbert B. Laney.

Good luck

is there any other example of the Riemann problem ?

A dam breaking in shallow water is one. Any problem governed by a hyperbolic equation has a Riemann problem (look up the theory of Riemann invariants in a book on pdes).

Do you think someone can see it in two-phase flows ? any advice will be very much appreciated.