Time-dependent and -independent Schrödinger Equation Why do we need both, the time-dependent and time-independent Schrödinger Equations? 
 A: We don't actually need the time-independent equation, but if the potential is constant through time, you can assume that the wave function is separable, $\Psi(x,t)=\psi(x)\xi(t)$ which will give the usual time-independent equation. This method greatly simplifies analytical solutions.
However, it's always possible to find a numerical solution for a time-independent potential using only the time-dependent eq., although not practical.
A: Your question implies that the time-dependent and time-independent equations are somehow different things, but while technically true this is misleading. The time-independent form is a simplification we can make when the potential does not change with time, so if we start with the time-dependent equation we get the time-independent form for free.
As for why we use the time-independent equation: speaking as an (ex) chemist we're frequently interested in calculating the energy of molecules that are not changing with time. The time-independence means we can use the time-independent Schrödinger equation and it's quicker and easier to use i.e. less CPU time and memory needed to do the calculation.
