Why continue to teach and use hypothesis testing (when confidence intervals are available)? Why continue to teach and use hypothesis testing (with all its difficult concepts and which are among the most statistical sins) for problems where there is an interval estimator (confidence, bootstrap, credibility or whatever)? What is the best explanation (if any) to be given to students? Only tradition? The views will be very welcome.
 A: In teaching Neyman Pearson hypothesis testing to early statistics students, I have often tried to locate it in its original setting: that of making decisions.  Then the infrastructure of type 1 and type 2 errors all makes sense, as does the idea that you might accept the null hypothesis.  
We have to make a decision, we think that the outcome of our decision can be improved by knowledge of a parameter, we only have an estimate of that parameter.  We still have to make a decision.  Then what is the best decision to make in the context of having an estimate of the parameter?  
It seems to me that in its original setting (making decisions in the face of uncertainty) the NP hypothesis test makes perfect sense.  See e.g. N & P 1933, particularly p. 291.
Neyman and Pearson. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character (1933) vol. 231 pp. 289-337
A: This is my personal opinion, so I'm not sure it properly qualifies as an answer.
Why should we teach hypothesis testing?
One very big reason, in short, is that, in all likelihood, in the time it takes you to read this sentence, hundreds, if not thousands (or millions) of hypothesis tests have been conducted within a 10ft radius of where you sit.
Your cell phone is definitely using a likelihood ratio test to decide whether or not it is within range of a base station. Your laptop's WiFi hardware is doing the same in communicating with your router.
The microwave you used to auto-reheat that two-day old piece of pizza used a hypothesis test to decide when your pizza was hot enough.
Your car's traction control system kicked in when you gave it too much gas on an icy road, or the tire-pressure warning system let you know that your rear passenger-side tire was abnormally low, and your headlights came on automatically at around 5:19pm as dusk was setting in.
Your iPad is rendering this page in landscape format based on (noisy) accelerometer readings.
Your credit card company shut off your card when "you" purchased a flat-screen TV at a Best Buy in Texas and a $2000 diamond ring at Zales in a Washington-state mall within a couple hours of buying lunch, gas, and a movie near your home in the Pittsburgh suburbs.
The hundreds of thousands of bits that were sent to render this webpage in your browser each individually underwent a hypothesis test to determine whether they were most likely a 0 or a 1 (in addition to some amazing error-correction).
Look to your right just a little bit at those "related" topics.
All of these things "happened" due to hypothesis tests. For many of these things some interval estimate of some parameter could be calculated. But, especially for automated industrial processes, the use and understanding of hypothesis testing is crucial.

On a more theoretical statistical level, the important concept of statistical power arises rather naturally from a decision-theoretic / hypothesis-testing framework. Plus, I believe "even" a pure mathematician can appreciate the beauty and simplicity of the Neyman–Pearson lemma and its proof.
This is not to say that hypothesis testing is taught, or understood, well. By and large, it's not. And, while I would agree that—particularly in the medical sciences—reporting of interval estimates along with effect sizes and notions of practical vs. statistical significance are almost universally preferable to any formal hypothesis test, this does not mean that hypothesis testing and the related concepts are not important and interesting in their own right.
A: Hypothesis testing is a useful way to frame a lot of questions: is the effect of a treatment zero or nonzero? The ability between statements such as these and a statistical model or procedure (including the construction of an interval estimator) is important for practitioners I think.
It also bears mentioning that a confidence interval (in the traditional sense) isn't inherently any less "sin-prone" than hypothesis testing - how many intro stats students know the real definition of a confidence interval?
Perhaps the problem isn't hypothesis testing or interval estimation as it is the classical versions of the same; the Bayesian formulation avoids these quite nicely.
A: I teach hypothesis tests for a number of reasons.  One is historical, that they'll have to understand a large body of prior research they read and understand the hypothesis testing point of view.  A second is that, even in modern times, it's still used by some researchers, often implicitly, when performing other kinds of statistical analyses.
But when I teach it, I teach it in the framework of model building, that these assumptions and estimates are parts of building models. That way it's relatively easy to switch to comparing more complex and theoretically interesting models. Research more often pits theories against one another rather than a theory versus nothing.
The sins of hypothesis testing are not inherent in the math, and proper use of those calculations.  Where they primarily lie is in over-reliance and misinterpretation.  If a vast majority of naïve researchers exclusively used interval estimation with no recognition of any of the relationships to these things we call hypotheses we might call that a sin.
A: I personally feel we would be better off without hypothesis tests.  The only place I can think of where hypothesis tests offer something unique and useful is in the area of multiple degree of freedom joint hypothesis tests.  Examples include ANOVA for comparing more than two groups, simultaneous tests combining main effects and interactions (tests of total effect), and simultaneous tests combining linear and nonlinear terms related to a continuous predictor (multiple d.f. test of association).  For simple things, interval estimation is easier, and much less likely to mislead than $P$-values.  As said so well in the classic paper Absence of evidence is not evidence of absence, a large $P$-value contains no information.  $P$-values only provide evidence against a hypothesis, never in favor of it (Fisher's response when asked how to interpret a large $P$-value was "Get more data").  A confidence or credible interval keeps the researcher more honest by describing how much she doesn't know.
A: The reason is decision making. In most decision making you either do it or not. You may keep looking at intervals all day long, in the end there's a moment where you decide to do it or not. 
Hypothesis testing fits nicely into this simple reality of YES/NO.
A: I think it depends on which hypothesis testing you are talking about.  The "classical" hypothesis testing (Neyman-Pearson) is said to be defective because it does not appropriately condition on what actually happened when you did the test.  It instead is designed to work "regardless" of what you actually saw in the long run.  But failing to condition can lead to misleading results in the individual case.  This is simply because the procedure "does not care" about the individual case, on the long run.
Hypothesis testing can be cast in the decision theoretical framework, which I think is a much better way to understand it.  You can restate the problem as two decisions:


*

*"I will act as if $H_0$ is true"

*"I will act as if $H_\mathrm{A}$ is true"


The decision framework is much easier to understand, because it clearly separates out the concepts of "what will you do?" and "what is the truth?" (via your prior information).
You could even apply "decision theory" (DT) to your question.  But in order to stop hypothesis testing, DT says you must have an alternative decision available to you.  So the question is: if hypothesis testing is abandoned, what is to take its place?  I can't think of an answer to this question.  I can only think of alternative ways to do hypothesis testing.
(NOTE: in the context of hypothesis testing, the data, sampling distribution, prior distribution, and loss function are all prior information because they are obtained prior to making the decision.)
A: If I were a hardcore Frequentist I would remind you that confidence intervals are quite regularly just inverted hypothesis tests, i.e. when the 95% interval is simply another way of describing all the points that a test involving your data wouldn't reject at the .05 level.  In these situations a preference for one over the other is question of exposition rather than method.
Now, exposition is important of course, but I think that would be a pretty good argument.  It's neat and clarifying to explain the two approaches as restatements of the same inference from different points of view.  (The fact that not all interval estimators are inverted tests is then an inelegant but not particularly awkward fact, pedagogically speaking).
Much more serious implications come from the decision to condition on the observations, as pointed out above.  However, even in retreat the Frequentist could always observe that there are plenty of situations (perhaps not a majority) where conditioning on the observations would be unwise or unilluminating.  For those, the HT/CI setup is (not 'are') exactly what is wanted, and should be taught as such.
