Phylogenies -- which is what cladistics is designed to determine -- are certainly of practical use for me. I use the genomes of closely related species to determine which alleles are ancestral and which are derived, for identifying cases of positive selection (among other uses); without knowing which species to compare, I would have no basis for extracting the information. We use phylogenies to determine which species to sequence so as to get the most bang for our sequencing bucks. We use them to determine which parts of the genome are functional, since they're the parts that are conserved across species. In particular, we need them to determine which parts are functional only within a particular lineage.
More broadly, without the framework of common descent and an implied phylogenetic tree, comparative genomics would have no coherence, and we'd have no structure for thinking about our data. I have no idea how we'd function without it.