We have,

$C_1:\left|z_1\right|=9$

$C_2:\left|z_2-3-4i\right|=4$

Centres of the circles are $c_1\equiv \left(0,0\right)$ and $c_2\equiv \left(3,4\right)$.

Radius of the circles are $r_1=9$ and $r_2=4$.

$\Rightarrow c_1{c}_2=\sqrt{3^2+4^2}=5$ and $r_1-r_2=5$

Circles touch each other internally.

$\Rightarrow {\left|z_1-z_2\right|}_{\min }=0$ at the point of contact.