The displacement of an oscillator is written as $y ( t ) = a \sin (ω t ± φ)$ whereas the equation of a wave is written as $y ( x , t ) = a \sin (ω t − kx )$.
The differences I can enumerate are-
The former, SHM is an equation of two dimensions. SHM is the function of time. And the former, travelling wave equation is a three dimensional equation- a function of displacement and time.
SHM depends upon the initial phase $φ$, whereas the wave equation depend upon $k$= wave number, and $x$= displacement.
The wave equation is the partial differentiated solution of the SHM.
I cannot find any other differences. Are there any other differences?