I have been thinking about resolution of singularities of plane curves in terms of blow-up and integral closure and i am trying to see how the two approaches relate.
Let us consider the affine curve given by the equation $y^2 - x^2 - x^3 = 0$. Then its blow-up is a parabola, which is nonsingular. This is explained in Hartshorne example I.1.4.9 (see also Example I.4.9.1 in Hartshorne (blowing-up)) or in Fulton's Algebraic Curves page 84.
On the other hand, the integral closure of $k[x,y]/(y^2 - x^2 - x^3)$ is $k[t]$ (see https://mathoverflow.net/questions/1504/what-is-the-geometric-meaning-of-integral-closure or isolating locally the branches of a curve) and so the affine variety that corresponds to the integral closure is just the affine line.
Question: The curve that we obtain from the blow-up is geometrically related to the original curve, i.e. the original geometry of the curve is respected "as much as possible". On the other hand, the affine line that we obtain via the integral closure has completely eliminated the original geometry. So what is the point of considering the integral closure? I mean, we do want to "resolve" the singularity at $(0,0)$ but the integral closure seems to be doing this in very "crude" way.