The chemical structure of the very young starless core L1521E

L1521E is a dense starless core in Taurus that was found to have relatively low molecular depletion by earlier studies, thus suggesting a recent formation. We aim to characterize the chemical structure of L1521E and compare it to the more evolved L1544 pre-stellar core. We have obtained $\sim$2.5$\times$2.5 arcminute maps toward L1521E using the IRAM-30m telescope in transitions of various species. We derived abundances for the species and compared them to those obtained toward L1544. We estimated CO depletion factors. Similarly to L1544, $c$-C$_3$H$_2$ and CH$_3$OH peak at different positions. Most species peak toward the $c$-C$_3$H$_2$ peak. The CO depletion factor derived toward the $Herschel$ dust peak is 4.3$\pm$1.6, which is about a factor of three lower than that toward L1544. The abundances of sulfur-bearing molecules are higher toward L1521E than toward L1544 by factors of $\sim$2-20. The abundance of methanol is similar toward the two cores. The higher abundances of sulfur-bearing species toward L1521E than toward L1544 suggest that significant sulfur depletion takes place during the dynamical evolution of dense cores, from the starless to pre-stellar stage. The CO depletion factor measured toward L1521E suggests that CO is more depleted than previously found. Similar CH$_3$OH abundances between L1521E and L1544 hint that methanol is forming at specific physical conditions in Taurus, characterized by densities of a few $\times$10$^4$ cm$^{-3}$ and $N$(H$_2$)$\gtrsim$10$^{22}$ cm$^{-2}$, when CO starts to catastrophically freeze-out, while water can still be significantly photodissociated, so that the surfaces of dust grains become rich in solid CO and CH$_3$OH, as already found toward L1544. Methanol can thus provide selective crucial information about the transition region between dense cores and the surrounding parent cloud.