See my other comment regarding energy eigenstates (stationary states). Most physical states are superpositions of stationary states, not stationary states themselves.

I suspect the version of the Schrodinger Equation you're talking about is the time-independent Schrodinger equation. All wavefunctions obey the time-dependent Schrodinger equation, but only stationary states obey the time-independent. The time independent Schrodinger equation is really just a convenient way to find the stationary states of a system. Stationary states have trivial time-dependence (all that happens is their phase rotates in time, while their magnitude remains constant). These are then used to build up the rest of the states of the system, most of which have non-trivial time dependence. A typical state in the real world would be a wavepacket, where the size of the packet spreads over time as it travels along.