Context. Most of our knowledge of the stellar component of galaxies is based on the analysis of distant systems and comes from integrated light data. It is important to test whether the results of the star formation histories (SFH) obtained with standard full-spectrum fitting methods are in agreement with those obtained through colour-magnitude diagram (CMD) fitting (usually considered the most reliable approach). Aims: We compare SFHs recovered from the two techniques in Leo A, a Local Group dwarf galaxy most of whose stars formed during the last 8 Gyrs. This complements our previous findings in a field in the Large Magellanic Cloud bar, where star formation has been in progress since early epochs at varying rates. Methods: We have used GTC/OSIRIS in long-slit mode to obtain a high-quality integrated light spectrum by scanning a selected region within Leo A, for which a CMD reaching the old main sequence turn-off (oMSTO) is available from HST. We compared the SFH obtained from the two datasets, using state-of-art methods of integrated light (STECKMAP) and resolved stellar population analysis. In the case of the CMD, we computed the SFH both from a deep CMD (observed with HST/ACS) and from a shallower CMD (archival data from HST/WFPC2). Results: The agreement between the SFHs recovered from the oMSTO CMD and from full spectrum fitting is remarkable, particularly regarding the time evolution of the star formation rate. The overall extremely low metallicity of Leo A is recovered up to the last 2 Gyrs when some discrepancies appear. A relatively high metallicity found for the youngest stars from the integrated data is a recurring feature that might indicate that the current models or synthesis codes should be revised, but that can be significantly mitigated using a more restrictive metallicity range. We thoroughly inspect the robustness of both approaches separately, finding that the subtle differences between them are inherent to the methods themselves. The SFH recovered from the shallow CMD also presents differences with the other two. Conclusions: Modern full-spectral fitting codes are able to recover both average constant SFHs (LMC case) and SFHs with a dominant fraction of young stellar populations. The analysis of high S/N spectra seems to provide more reliable SFH estimates than that of CMDs not reaching the oMSTO. The comparison presented in this paper needs to be repeated for predominantly old systems, thus assessing the performance of full-spectrum fitting for a full range of SFHs.