Think your statement through as a Frequentist and make it more specific first. A Frequentist could not say that "data set A is different from data set B", without any further clarification.
First, you'd have to state what you mean by "different". Perhaps you mean "have different mean values". Then again, you might mean "have different variances". Or perhaps something else?
Then, you'd have to state what kind of test you would use, which depends on what you believe are valid assumptions about the data. Do you assume that the data sets are both normally-distributed about some means? Or do you believe that they are both Beta-distributed? Or something else?
Now can you see that the second decision is much like the priors in Bayesian statistics? It's not just "my past experience", but is rather what I believe, and what I believe my peers will believe, are reasonable assumptions about my data. (And Bayesians can use uniform priors, which pushes things towards Frequentist calculations.)
EDIT: In response to your comment: the next step is contained in the first decision I mentioned. If you want to decide whether the means of two groups are different, you would look at the distribution of the difference of the means of the two groups to see if this distribution does or does not contain zero, at some level of confidence. Exactly how close to zero you count as zero and exactly which portion of the (posterior) distribution you use are determined by you and the level of confidence you desire.
A discussion of these ideas can be found in a paper by Kruschke, who also wrote a very readable book Doing Bayesian Data Analysis, which covers an example on pages 307-309, "Are Different Groups Equal?". (Second edition: p. 468-472.) He also has a blog posting on the subject, with some Q&A.
FURTHER EDIT: Your description of the Bayesian process is also not quite correct. Bayesians only care about what the data tells us, in light of what we knew independent of the data. (As Kruschke points out, the prior does not necessarily occur before the data. That's what the phrase implies, but it's really just our knowledge excluding some of the data.) What we knew independently of a particular set of data may be vague or specific and may be based on consensus, a model of the underlying data generation process, or may just be the results of another (not necessarily prior) experiment.
The cited article seems to be based on fears that statisticians "will not be an intrinsic part of the scientific team, and the scientists will naturally have their doubts about the methods used" and that "collaborators will view us as technicians they can steer to get their scientific results published." My comments on the questions posed by @rvl come from the perspective of a non-statistician biological scientist who has been forced to grapple with increasingly complicated statistical issues as I moved from bench research to translational/clinical research over the past few years. Question 5 is clearly answered by the multiple answers now on this page; I'll go in reverse order from there.
4) It doesn't really matter whether an "exact model" exists, because even if it does I probably won't be able to afford to do the study. Consider this issue in the context of the discussion: Do we really need to include “all relevant predictors?” Even if we can identify "all relevant predictors" there will still be the problem of collecting enough data to provide the degrees of freedom to incorporate them all reliably into the model. That's hard enough in controlled experimental studies, let alone retrospective or population studies. Maybe in some types of "Big Data" that's less of a problem, but it is for me and my colleagues. There will always be the need to "be smart about it," as @Aksakal put it an an answer on that page.
In fairness to Prof. van der Laan, he doesn't use the word "exact" in the cited article, at least in the version presently available on line from the link. He talks about "realistic" models. That's an important distinction.
Then again, Prof. van der Laan complains that "Statistics is now an art, not a science," which is more than a bit unfair on his part. Consider the way he proposes to work with collaborators:
... we need to take the data, our identity as a statistician, and our scientific collaborators seriously. We need to learn as much as possible about how the data were generated. Once we have posed a realistic statistical model, we need to extract from our collaborators what estimand best represents the answer to their scientific question of interest. This is a lot of work. It is difficult. It requires a reasonable understanding of statistical theory. It is a worthy academic enterprise!
The application of these scientific principles to real-world problems would seem to require a good deal of "art," as with work in any scientific enterprise. I've known some very successful scientists, many more who did OK, and some failures. In my experience the difference seems to be in the "art" of pursing scientific goals. The result might be science, but the process is something more.
3) Again, part of the issue is terminological; there's a big difference between an "exact" model and the "realistic" models that Prof. van der Laan seeks. His claim is that many standard statistical models are sufficiently unrealistic to produce "unreliable" results. In particular: "Estimators of an estimand defined in an honest statistical model cannot be sensibly estimated based on parametric models." Those are matters for testing, not opinion.
His own work clearly recognizes that exact models aren't always possible. Consider this manuscript on targeted maximum likelihood estimators (TMLE) in the context of missing outcome variables. It's based on an assumption of outcomes missing at random, which may never be testable in practice: "...we assume there are no unobserved confounders of the relationship between missingness ... and the outcome." This is another example of the difficulty in including "all relevant predictors." A strength of TMLE, however, is that it does seem to help evaluate the "positivity assumption" of adequate support in the data for estimating the target parameter in this context. The goal is to come as close as possible to a realistic model of the data.
2) TMLE has been discussed on Cross Validated previously. I'm not aware of widespread use on real data. Google Scholar showed today 258 citations of what seems to be the initial report, but at first glance none seemed to be on large real-world data sets. The Journal of Statistical Software article on the associated R package only shows 27 Google Scholar citations today. That should not, however, be taken as evidence about the value of TMLE. Its focus on obtaining reliable unbiased estimates of the actual "estimand" of interest, often a problem with plug-in estimates derived from standard statistical models, does seem potentially valuable.
1) The statement: "a statistical model that makes no assumptions is always true" seems to be intended as a straw man, a tautology. The data are the data. I assume that there are laws of the universe that remain consistent from day to day. The TMLE method presumably contains assumptions about convexity in the search space, and as noted above its application in a particular context might require additional assumptions.
Even Prof. van der Laan would agree that some assumptions are necessary. My sense is that he would like to minimize the number of assumptions and to avoid those that are unrealistic. Whether that truly requires giving up on parametric models, as he seems to claim, is the crucial question.
Best Answer
I wouldn't consider non-parametric or robust as being sub-categories of statistics in the way that frequentist and Bayesian are, simply because there are both frequentist and Bayesian methods for non-parametric and robust statistics. Frequentist and Bayesian are genuine sub-categories as they are based on fundamentally different definitions of a probability. Frequentists and Bayesians will both vary the strength of assumptions made depending on the requirements of the application.
So I would say that particular subdivision into four categories is not widely recognised in statistics. In my opinion, both Bayesian and frequentist methods can be used for most statistical problems, however they are not always equally useful, for example whether a frequentist confidence interval or a Bayesian credible interval is more appropriate depends on whether you want to ask a question about what to expect if the experiment were replicated, or what we can conclude about the statistics as a result of the particular experiment that we have actually performed (I would suggest in most cases it is the latter, but scientists generally use frequentist methods anyway).