When solving business problems using data, it's common that at least one key assumption that under-pins classical statistics is invalid. Most of the time, no one bothers to check those assumptions so you never actually know.
For instance, that so many of the common web metrics are "long-tailed" (relative to the normal distribution) is, by now, so well documented that we take it for granted. Another example, online communities--even in communities with thousands of members, it's well-documented that by far the largest share of contribution to/participation in many of these community is attributable to a minuscule group of 'super-contributors.' (E.g., a few months ago, just after the SO API was made available in beta, a StackOverflow member published a brief analysis from data he collected through the API; his conclusion--less than one percent of the SO members account for most of the activity on SO (presumably asking questions, and answering them), another 1-2% accounted for the rest, and the overwhelming majority of the members do nothing).
Distributions of that sort--again more often the rule rather than the exception--are often best modeled with a power law density function. For these type of distributions, even the central limit theorem is problematic to apply.
So given the abundance of populations like this of interest to analysts, and given that classical models perform demonstrably poorly on these data, and given that robust and resistant methods have been around for a while (at least 20 years, I believe)--why are they not used more often? (I am also wondering why I don't use them more often, but that's not really a question for CrossValidated.)
Yes I know that there are textbook chapters devoted entirely to robust statistics and I know there are (a few) R Packages (robustbase is the one I am familiar with and use), etc.
And yet given the obvious advantages of these techniques, they are often clearly the better tools for the job--why are they not used much more often? Shouldn't we expect to see robust (and resistant) statistics used far more often (perhaps even presumptively) compared with the classical analogs?
The only substantive (i.e., technical) explanation I have heard is that robust techniques (likewise for resistant methods) lack the power/sensitivity of classical techniques. I don't know if this is indeed true in some cases, but I do know it is not true in many cases.
A final word of preemption: yes I know this question does not have a single demonstrably correct answer; very few questions on this Site do. Moreover, this question is a genuine inquiry; it's not a pretext to advance a point of view--I don't have a point of view here, just a question for which i am hoping for some insightful answers.

Researchers want small p-values, and you can get smaller p-values if you use methods that make stronger distributional assumptions. In other words, non-robust methods let you publish more papers. Of course more of these papers may be false positives, but a publication is a publication.  That's a cynical explanation, but it's sometimes valid.