Context. 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. Aims. We aim to characterize the chemical structure of L1521E and compare it to the more evolved L1544 pre-stellar core. Methods. We have obtained ~2.5 × 2.5 arcminute maps toward L1521E using the IRAM-30 m telescope in transitions of various species, including C17O, CH3OH, c-C3H2, CN, SO, H2CS, and CH3CCH. We derived abundances for the observed species and compared them to those obtained toward L1544. We estimated CO depletion factors using the C17O IRAM-30 m map, an N(H2) map derived from Herschel/SPIRE data and a 1.2 mm dust continuum emission map obtained with the IRAM-30 m telescope. Results. Similarly to L1544, c-C3H2 and CH3OH peak at different positions. Most species peak toward the c-C3H2 peak including C2S, C3S, HCS+, HC3N, H2CS, CH3CCH, and C34S. C17O and SO peak close to both the c-C3H2 and the CH3OH peaks. CN and N2H+ peak close to the Herschel dust peak. We found evidence of CO depletion toward L1521E. The lower limit of the CO depletion factor derived toward the Herschel dust peak is 4.3±1.6, which is about a factor of three lower than toward L1544. We derived abundances for several species toward the dust peaks of L1521E and L1544. The abundances of most sulfur-bearing molecules such as C2S, HCS+, C34S, C33S, and HCS+ are higher toward L1521E than toward L1544 by factors of ~2–20, compared to the abundance of A-CH3OH. The abundance of methanol is very similar toward the two cores. Conclusions. The fact that the abundances of sulfur-bearing species toward L1521E are higher than toward L1544 suggests 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 CH3OH abundances between L1521E and L1544 hint that methanol is forming at specific physical conditions in the Taurus Molecular Cloud Complex, characterized by densities of a few ×104 cm−3 and N(H2) ≳ 1022 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 CH3OH, as already found toward L1544. Methanol can thus provide selective crucial information about the transition region between dense cores and the surrounding parent cloud.