While studies of gas-phase metallicity gradients in disc galaxies are common, very little has been done towards the acquisition of stellar abundance gradients in the same regions. We present here a comparative study of the stellar metallicity and age distributions in a sample of 62 nearly face-on, spiral galaxies with and without bars, using data from the CALIFA survey. We measure the slopes of the gradients and study their relation with other properties of the galaxies. We find that the mean stellar age and metallicity gradients in the disc are shallow and negative. Furthermore, when normalized to the effective radius of the disc, the slope of the stellar population gradients does not correlate with the mass or with the morphological type of the galaxies. In contrast to this, the values of both age and metallicity at ~2.5 scale lengths correlate with the central velocity dispersion in a similar manner to the central values of the bulges, although bulges show, on average, older ages and higher metallicities than the discs. One of the goals of the present paper is to test the theoretical prediction that non-linear coupling between the bar and the spiral arms is an efficient mechanism for producing radial migrations across significant distances within discs. The process of radial migration should flatten the stellar metallicity gradient with time and, therefore, we would expect flatter stellar metallicity gradients in barred galaxies. However, we do not find any difference in the metallicity or age gradients between galaxies with and without bars. We discuss possible scenarios that can lead to this lack of difference. Tables 1-3 and Appendices are available in electronic form at http://www.aanda.org