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The Type Ia supernovae (SNe Ia) 2011by, hosted in NGC 3972, and 2011fe, hosted in M101, are optical ‘twins,’ having almost identical optical light-curve shapes, colours, and near-maximum-brightness spectra. However, SN 2011fe had significantly more ultraviolet (UV; 1600 < λ < 2500 Å) flux than SN 2011by before and at peak luminosity. Several theoretical models predict that SNe Ia with higher progenitor metallicity should (1) have additional UV opacity and thus lower UV flux; (2) have an essentially unchanged optical spectral-energy distribution; (3) have a similar optical light-curve shape; and (4) because of the excess neutrons, produce more stable Fe-group elements at the expense of radioactive 56Ni and thus have a lower peak luminosity. Following these predictions, Foley and Kirshner suggested that the difference in UV flux between SNe 2011by and 2011fe was the result of their progenitors having significantly different metallicities. They also measured a large, but insignificant, difference between the peak absolute magnitudes of the SNe (ΔMV, peak = 0.60 ± 0.36 mag), with SN 2011fe being more luminous. We present a new Cepheid-based distance to NGC 3972, substantially improving the precision of the distance measurement for SN 2011by. With these new data, we determine that the SNe have significantly different peak luminosities (ΔMV, peak = 0.335 ± 0.069 mag). Consequently, SN 2011fe produced 38 per cent more 56Ni than SN 2011by, consistent with predictions for progenitor metallicity differences for these SNe, although alternative models may also explain this difference. We discuss how progenitor metallicity differences can contribute to the intrinsic scatter for light-curve-shape-corrected SN luminosities, the use of ‘twin’ SNe for measuring distances, and implications for using SNe Ia for constraining cosmological parameters.