In this paper, we deal with codes obtained from Butson-Hadamard matrices, called BH codes, focusing on their minimum distances. We first consider the usual Hamming distance and find lower bounds for distances of BH codes. Then we turn our attention to homogeneous weights, and search for distances of BH code families under these weights. Next, we introduce the notion of quasi-homogeneous weights as a generalization of homogeneous weights and show that certain BH codes equipped with quasi-homogeneous weights are Plotkin optimal. In addition, we obtain distances of BH codes under certain quasi-homogeneous weights. Our results are applied to determine parameters of p-ary codes projected under Gray isometries from BH codes over DOUBLE-STRUCK CAPITAL Zpe, where p is a prime number and e >= 2 is an integer.