Let us observe some simpler case: $(\sqrt{x})^2$ and $\sqrt{(x)^2}$.
For first expression we have to find root of $x$ first and then square that result. Since the square root is defined only for nonnegative real numbers first domain is $x\in [0,+\infty)$ and $(\sqrt{x})^2 = x$.
For second expression we have to square $x$ first and then find root of that result. Since square is defined for all real numbers and always gives nonnegative result second domain is $x\in(-\infty,+\infty)$ and $\sqrt{(x)^2} = |x|$.
Now we can apply same reasoning on your specific case and result is:
for $\left(\sqrt{(6x-7)}\right)^2$, domain is $[\frac{7}{6},+\infty)$ and the expression is $6x-7$.
for $\sqrt{(6x-7)^2}$, domain is $x\in (-\infty,+\infty)$ and expression is $|6x-7|$.