We compare the stellar population properties in the central regions of visually classified non-star-forming spiral and elliptical galaxies from Galaxy Zoo and Sloan Digital Sky Survey (SDSS) Data Release 7. The galaxies lie in the redshift range 0.04 < z < 0.1 and have stellar masses larger than log M* = 10.4. We select only face-on spiral galaxies in order to avoid contamination by light from the disc in the SDSS fibre and enabling the robust visual identification of spiral structure. Overall, we find that galaxies with larger central stellar velocity dispersions, regardless of morphological type, have older ages, higher metallicities and an increased overabundance of α-elements. Age and α-enhancement, at fixed velocity dispersion, do not depend on morphological type. The only parameter that, at a given velocity dispersion, correlates with morphological type is metallicity, where the metallicity of the bulges of spiral galaxies is 0.07 dex higher than that of the ellipticals. However, for galaxies with a given total stellar mass, this dependence on morphology disappears. Under the assumption that, for our sample, the velocity dispersion traces the mass of the bulge alone, as opposed to the total mass (bulge+disc) of the galaxy, our results imply that the formation epoch of galaxy and the duration of its star-forming period are linked to the mass of the bulge. The extent to which metals are retained within the galaxy, and not removed as a result of outflows, is determined by the total mass of the galaxy.