Despite the fact that Hartshorne (in Algebraic Geometry, Ch. 5, Exercise 3.8) asserts that the singularities of the plane curves
$$x^4-xy^4=0$$ and $$x^4-x^2y^3-x^2y^5+y^8=0,$$ have "the same configuration of infinitely near singular points with the same multiplicities", it seems to me that after a single blowup the new first curve possesses a single singular point of multiplicity $2$, while the new second curve possesses a single singular point of multiplicity $3$; who (if anyone) is correct?