Relation between continuous time transfer function and sampled approximation
$H(jω)$, which is DTFT of the transfer function $h(n)$, is basically a special case of the Z transform $H(z)$, where $z=e^jω$ i.e. $|z|=1$.
In this special case, the unit circle ($|z|=1$) should be included in the region of convergence (ROC) of $H(z)$. In other words, the system (which is LTI in your example) should be BIBO stable.
The system in your example is BIBO stable since it has one pole in $z=0$, therefore its ROC ($z≠0$) includes the unit circle. Hence, we can say that $H(z)=H(jw)$ only when $|z|=1$ i.e. on the unit circle.