As usual, a rant from Joe Felsenstein's webpage seems relevant. Specifically, this:
"There is no consensus, even among systematists, as to what the word "cladistics" means."
Everyone seems to have agreed that phenetics--which is to say the blind faith in the results of clustering algorithms such as neighbor-joining etc.--is on its own not very useful, compared to a principled dedication to placing organisms taxonomically into monophyletic groups.
However, the hard-line cladistics point of view (e.g. Willi Hennig society) is that no method of inference other than maximum parsimony (in one of its various flavors) is acceptable, for obscure philosophical reasons that no one cares about. Frankly, for any realistic purpose, cladistically unacceptable methods such as Bayesian inference and likelihood methods perform the best. From cladistics wiki:
Although traditionally such cladograms were generated largely on the basis of morphological characters and originally calculated by hand, genetic sequencing data and computational phylogenetics are now commonly used in phylogenetic analyses, and the parsimony criterion has been abandoned by many phylogeneticists in favor of more "sophisticated" but less parsimonious evolutionary models of character state transformation. Cladists contend that these models are unjustified.
In other words, cladistics is more of a philosophical point of view than a group of methods. There are "soft" versions that everyone pretty much agrees on, namely the principle that monophyletic groups are preferable to paraphyletic groups. But methodologically, it's just a lot easier and more workable to use non-cladistic or even phenetic techniques rather than the (computationally frequently more onerous and not necessarily more accurate) parsimony techniques of cladistics.
For a (fairly) recent apologia specific to neighbor-joining, see here.
In short, given the highly heuristic nature of most science, most workers aren't really paying attention to these debates, and just use the tool that works for the problem they're working on. And in many cases those methods are phenetic, even if cladists claimed a victory of principle.