Both the incompressibility Ka of a finite nucleus of mass A and that ($K_{infty}$) of infinite nuclear matter are fundamentally important for many critical issues in nuclear physics and astrophysics. While some consensus has been reached about the $K_{infty}$, accurate theoretical predictions and experimental extractions of $K_{tau}$ characterizing the isospin dependence of Ka have been very difficult. We propose a differential approach to extract the Kt and Ki independently from the Ka data of any two nuclei in a given isotope chain. Applying this new method to the Ka data from isoscalar giant monopole resonances (ISGMR) in even-even Pb, Sn, Cd and Ca isotopes taken by U. Garg {it et al.} at the Research Center for Nuclear Physics (RCNP), Osaka University, Japan, we find that the $^{106}$Cd-$^{116}$Cd and $^{112}$Sn-$^{124}$Sn pairs having the largest differences in isospin asymmetries in their respective isotope chains measured so far provide consistently the most accurate up-to-date Kt value of $K_{tau}=-616pm 59$ MeV and $K_{tau}=-623pm 86$ MeV, respectively, largely independent of the remaining uncertainties of the surface and Coulomb terms in expanding the $K_{rm A}$, while the $K_{infty}$ values extracted from different isotopes chains are all well within the current uncertainty range of the community consensus for $K_{infty}$. Moreover, the size and origin of the Soft Sn Puzzle is studied with respect to the Stiff Pb Phenomenon. It is found that the latter is favored due to a much larger (by $sim 380$ MeV) Kt for Pb isotopes than for Sn isotopes, while the Ki from analyzing the Ka data of Sn isotopes is only about 5 MeV less than that from analyzing the Pb data.
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