LTI system impulse response output
If your input signal is

$$x[n]=u[n]-u[n-1]$$

then it is in fact a unit impulse: $x[n]=\delta[n]$, which has only one non-zero value for $n=0$ ($x[0]=1$). All other values are zero. So the response to this input signal is the system's impulse response, because that's the definition of the impulse response: $y[n]=h[n]$

Note that $u[n]$ is the unit step, and the response to a unit step is not the impulse response but the step response. But for this example you don't need to compute the step response, because, as already mentioned, the input signal is simply a unit impulse.