We identify 66 photometrically classified Type Ia supernovae (SNe Ia) from the Dark Energy Survey (DES) that have occurred within red-sequence selected galaxy clusters. We compare light-curve and host galaxy properties of the cluster SNe to 1024 DES SNe Ia located in field galaxies, the largest comparison of two such samples at high redshift (z > 0.1). We find that cluster SN light curves decline faster than those in the field (97.7 per cent confidence). However, when limiting these samples to host galaxies of similar colour and mass, there is no significant difference in the SN light-curve properties. Motivated by previous detections of a higher-normalized SN Ia delay-time distribution in galaxy clusters, we measure the intrinsic rate of SNe Ia in cluster and field environments. We find the average ratio of the SN Ia rate per galaxy between high-mass ($10\le \log \mathrm{(\mathit{ M}_{*}/{\rm \mathit{ M}}_{\odot })} \le 11.25$) cluster and field galaxies to be 0.594 ± 0.068. This difference is mass-dependent, with the ratio declining with increasing mass, which suggests that the stellar populations in cluster hosts are older than those in field hosts. We show that the mass-normalized rate (or SNe per unit mass) in massive-passive galaxies is consistent between cluster and field environments. Additionally, both of these rates are consistent with rates previously measured in clusters at similar redshifts. We conclude that in massive-passive galaxies, which are the dominant hosts of cluster SNe, the cluster delay-time distribution is comparable to the field.