If we are studying a physical system which has some symmetries, why do we calculate conserved charges? What do the conserved charges tell about the system that the symmetries do not?
They help you solve dynamical physical problems easily.
Conserved charges arising from Noether currents give you different conservation laws - like conservation of angular momentum, linear momentum, energy, electric charge, etc. If you know that your physical setup must obey these certain conservation laws, you can then use them as tools to solve problems that involve physical phenomena in the setup. For example, you can reduce the number of unknown variables in a physical problem by using a conservation law, and then completely determine the dynamics of the system - I'm sure you've used the law of conservation of momentum in a collision to deduce final momenta from (given) initial momenta.
Because nature has certain symmetries, it helps us to determine unknown variables by simple applications of the ensuing conservation laws.