An interesting example where the resultant and the "reduced resultant" differ comes from the theory of elliptic curves. Take an elliptic curve $$E:\qquad y^2=x^3+ax+b$$ where $a$ and $b$ are integers. The duplication formula for $E$ states that on the elliptic curve $[2] (x_1,y_1)=(x_2,y_2)$ where $x_2=g(x_1)/4f(x_1)$, $$f(x)=x^3+ax+b\qquad{\rm and}\qquad g(x)=x^4-2ax^2-8bx+a^2.$$ The resultant of $f$ and $g$ is $(4a^3+27b^2)^2$ (the square of the discriminant of $E$ so nonzero). But the reduced discriminant is $|4a^3+27b^2|$ which one sees by noting that this divides all entries of the adjugate of the "resultant matrix" of $f$ and $g$.

The fact that this resultant is nonzero is used in a standard proof of the inequality $$h([2] P)\ge 4 h(P)-O(1)$$ for the naive height on $E$ (see Silverman's book).

An interesting example where the resultant and the "reduced resultant" differ comes from the theory of elliptic curves. Take an elliptic curve $$E:\qquad y^2=x^3+ax+b$$ where $a$ and $b$ are integers. The duplication formula for $E$ states that on the elliptic curve $[2] (x_1,y_1)=(x_2,y_2)$ where $x_2=g(x_1)/4f(x_1)$, $$f(x)=x^3+ax+b\qquad{\rm and}\qquad g(x)=x^4-2ax^2-8bx+a^2.$$ The resultant of $f$ and $g$ is $(4a^3+27b^2)^2$ (the square of the discriminant of $E$ so nonzero). But the reduced resultant is $|4a^3+27b^2|$ which one sees by noting that this divides all entries of the adjugate of the "resultant matrix" of $f$ and $g$.

The fact that this resultant is nonzero is used in a standard proof of the inequality $$h([2] P)\ge 4 h(P)-O(1)$$ for the naive height on $E$ (see Silverman's book).