You might want to read up on the principal specialization.

It specializes a symmetric function into a formal power series. For example, the symmetric function $s_1(x) = x_1+x_2+ \dotsb$ has principal specialization $s_1(1,q,q^2,\dotsc) = 1+q+q^2+\dotsb = \frac{1}{1-q}$, so it does not make sense to let $q\to 1$. The expression only exist as a formal power series. However, you always compute the finite version, $s_\lambda(1,q,q^2,\dotsc,q^{n})$, which is gonna be a polynomial in $q$.

So, the difference between your Jacob-Trudi formula and the cited q-formula, is the number of variables (finite vs, infinite).

The principal specialization is in many instances nicer than the finite number of variables specialization.

You might want to read up on the principal specialization.

It specializes a symmetric function into a formal power series. For example, the symmetric function $s_1(x) = x_1+x_2+ \dotsb$ has principal specialization $s_1(1,q,q^2,\dotsc) = 1+q+q^2+\dotsb = \frac{1}{1-q}$, so it does not make sense to let $q\to 1$. The expression only exist as a formal power series. However, you always compute the finite version, $s_\lambda(1,q,q^2,\dotsc,q^{n})$, which is gonna be a polynomial in $q$.

So, the difference between your Jacob-Trudi formula and the cited q-formula, is the number of variables (finite vs, infinite).

The principal specialization is in many instances nicer than the finite number of variables specialization.