[Abridged] We use optical IFS of nearby SN host galaxies provided by the CALIFA Survey with the goal of finding correlations in the environmental parameters at the location of different SN types. We recover the sequence in association of different SN types to the star-forming regions by using several indicators of the ongoing and recent SF related to both the ionized gas and the stellar populations. While the total ongoing SF is on average the same for the three SN types, SNe Ibc/IIb tend to happen closer to star-forming regions and occur in higher SF density locations compared to SNe II and SNe~Ia, the latter showing the weakest correlation. SNe~Ia host galaxies have on average masses that are $∼$0.3-0.8~dex higher than CC SNe hosts due to a larger fraction of old stellar populations in the SNe~Ia hosts. Using the recent SN~Ia delay-time distribution and the SFHs of the galaxies, we show that the SN~Ia hosts in our sample should presently produce a factor 2 more SNe~Ia than the CC~SN hosts. Since both types are in hosts with similar SF rate and hence similar CC~SN rate, this can explain the mass difference between the SN~Ia and CC~SN hosts, and reinforce the finding that at least part of SNe~Ia should originate from very old progenitors. Comparing the mean SFH of the eight least massive galaxies to that of the massive SF SN~Ia hosts we find that the low-mass galaxies formed their stars over more extended time than the massive SN~Ia hosts. We estimate that the low-mass galaxies should produce by a factor of 10 less SNe~Ia, and a factor of 3 less CC~SNe than the high-mass group. Therefore the ratio between the number of CC~SNe and SNe~Ia is expected to increase with decreasing the galaxy mass. CC~SNe tend to explode at positions with younger stellar populations than the galaxy average, but the galaxy properties at SNe~Ia locations are one average the same as the global ones. ; Comment: (Accepted in A&A, 24 pages, 19 Figures, 9 Tables)