The fundamental differences between static electric field and magnetic field are the following:
(1) The sources, “e-particle Q” vs. “e-current Qv”;
(2) The way the fields induced by source, “divE” vs. “curlB”;
(3) The nature of the fields, “vector field E”vs. “axial vector field B”;
(4) Effects...
i think it is a great question, since it leads to an deeper question that, therefore, the magnetic field law should be able to be derived mathematically from the Coulomb law and velocity, nothing more.