We propose a Bayesian approach for estimating branching tree mixture models to compare drug-resistance pathways (i.e. patterns of sequential acquisition of resistance to individual antibiotics) that are observed among Mycobacterium tuberculosis isolates collected from treatment-naïve and treatment-experienced patients. Resistant pathogens collected from treatment-naïve patients are strains for which fitness costs of resistance were not sufficient to prevent transmission, whereas those collected from treatment-experienced patients reflect both transmitted and acquired resistance, the latter of which may or may not be associated with lower transmissibility. The comparison of the resistance pathways constructed from these two groups of drug-resistant strains provides insight into which pathways preferentially lead to the development of multiple drug resistant strains that are transmissible. We apply the proposed statistical methods to data from worldwide surveillance of drug-resistant tuberculosis collected by the World Health Organization over 13 years.