State‐of‐the‐art distributed‐memory computer clusters contain multicore CPUs with 16 and more cores. The second generation of the Intel Xeon Phi many‐core processor has more than 60 cores with 16 GB of high‐performance on‐chip memory. We contrast the performance of the second‐generation Intel Xeon Phi, code‐named Knights Landing (KNL), with 68 computational cores to the latest multicore CPU Intel Skylake with 18 cores. A special‐purpose code solving a system of nonlinear reaction‐diffusion partial differential equations with several thousands of point sources modeled mathematically by Dirac delta distributions serves as realistic test bed. The system is discretized in space by the finite volume method and advanced by fully implicit time‐stepping, with a matrix‐free implementation that allows the complex model to have an extremely small memory footprint. The sample application is a seven variable model of calcium‐induced calcium release (CICR) that models the interplay between electrical excitation, calcium signaling, and mechanical contraction in a heart cell. The results demonstrate that excellent parallel scalability is possible on both hardware platforms, but that modern multicore CPUs outperform the specialized many‐core Intel Xeon Phi KNL architecture for a large class of problems such as systems of parabolic partial differential equations.