Yes, there is a strong relationship between the two.

First, let's work locally in affine space rather than in projective space (it makes more sense to work locally just because we are dealing with a sheaf, which is defined locally). So I will consider a non-homogen

Working without a metric (as one does in at least the algebraic aspects of algebraic geometry), it is perhaps better to talk not about the gradient of $f$, but its exterior derivative $df$, given by the same formula: $df = f_x dx + f_y dy.$ Since this is differential form valued, we will compare it with the conormal bundle to the curve $C$ cut out by $f = 0$.

Now the exterior derivative can be thought of simply as taking the leading (i.e. linear) term of $f$.

On the other hand, if $\mathcal I$ is the ideal sheaf cutting out the curve $C$, then the conormal bundle is $\mathcal I/\mathcal I^2$. (If $f$ is degree $d$, then $\mathcal I = \mathcal O(-d)$, and so this can be rewritten as $\mathcal O(-d)\_{| C}$, dual to the normal bundle $\mathcal O(d)\_{| C}$.) Now $f$ is a section of $\mathcal I/\mathcal I^2$ (over the affine patch on which we are working), so we may certainly regard it as a section of $\mathcal I/\mathcal I^2$; this section is the (image in the conormal bundle to $C$ of) the exterior derivative of $f$.

The formula $\mathcal I/\mathcal I^2$ for the conormal bundle is thus simply a structural interpretation of the idea that we compute the normal to curve by taking the leading term of an equation for the curve.

Yes, there is a strong relationship between the two.

First, let's work locally in affine space rather than in projective space (it makes more sense to work locally just because we are dealing with a sheaf, which is defined locally). So I will consider a non-homogen

Working without a metric (as one does in at least the algebraic aspects of algebraic geometry), it is perhaps better to talk not about the gradient of $f$, but its exterior derivative $df$, given by the same formula: $df = f_x dx + f_y dy.$ Since this is differential form valued, we will compare it with the conormal bundle to the curve $C$ cut out by $f = 0$.

Now the exterior derivative can be thought of simply as taking the leading (i.e. linear) term of $f$.

On the other hand, if $\mathcal I$ is the ideal sheaf cutting out the curve $C$, then the conormal bundle is $\mathcal I/\mathcal I^2$. (If $f$ is degree $d$, then $\mathcal I = \mathcal O(-d)$, and so this can be rewritten as $\mathcal O(-d)\_{| C}$, dual to the normal bundle $\mathcal O(d)\_{| C}$.) Now $f$ is a section of $\mathcal I/\mathcal I^2$ (over the affine patch on which we are working), so we may certainly regard it as a section of $\mathcal I/\mathcal I^2$; this section is the (image in the conormal bundle to $C$ of) the exterior derivative of $f$.

The formula $\mathcal I/\mathcal I^2$ for the conormal bundle is thus simply a structural interpretation of the idea that we compute the normal to the curve by taking the leading term of an equation for the curve.