Why do we always characterize a LTI system by its impulse response?
Let's try a simple example.  A discrete-time LTI system has unit pulse response
$$h[0] = 1,\quad h[1] = 1,\quad h[n] = 0 ~~\text{for all}~ n \geq 2.
\tag{1}$$
The response $y$ of this system to a signal $x$ given by
$$x[0] = 1,\quad  x[1] = -1,\quad x[2] = 1,\quad x[n] = 0 
~~\text{for all}~ n \geq 3 \tag{2}$$ is readily calculated 
via the convolution formula,
and we get
$$y[0] = 1, \quad y[1] = 0, \quad y[2] = 0, \quad y[3] = 1,\quad y[n] = 0 
~~\text{for all}~ n \geq  4.\tag{3}$$

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The unit-step response $g$ of the same LTI system is easily found to be
$$g[0] = 1,\quad g[n] = 2 ~~\text{for all}~ n \geq 1.\tag{4}$$

So, suppose that you are given only the step response $g$ in $(4)$
but not the pulse response in $(1)$.  How will you compute the response
of this LTI system to input $x$?  It is known that the answer is $(3)$,
but you are asked to figure out how to get from $(2)$ and $(4)$ to $(3)$.