Why do Experimental Design when you could build classifiers? Say that you wanted to test whether people who took a placebo were more likely to smoke.
You can devise a experimental design/test for this, but I could also create a model that predicts whether someone will smoke given they took the placebo. If there is high enough correlation, I could determine that the change was significant.
What is the difference between constructing a model vs hypothesis testing?
 A: The key aspect of an experimental design is that the participants are randomly assigned to the treatment or the control. That way you can control for potential confounders without having to measure them. In a model you have to hope that all relevant confounders are in your data, and if they are, that they are correctly measured, which is never the case in real data.
A: 
What is the difference between constructing a model vs hypothesis testing?

Like @mkt mentioned - both of these involve models in one for or another so that's a false distinction between them.
Differences
As far as I am aware classification is related to separating classes based on some set of common features. It makes a mapping X -> Y where Y is a set of possible classes and X is the feature set.
And hypothesis testing on the other hand is typically concerned with point estimates, without assigning them into some kind of groups. More precisely it tries to estimate the probability to obtain the observed (or a more extreme than observed) point estimate if some null model (called the null hypotheses) is true. And that is called p-value.
More on differences
Caltech Professor Yaser Abu-Mostafa in his online course "learning from data" defines these as the key elements for constructing classifiers:


*

*There is a patter to "learn".

*The pattern cannot be written down mathematically.

*We have data to train on.


The thing is that once we have a concrete null hypothesis in mind the problem becomes mathematical - there is no learning required. We have a null model for generating the data and a procedure to get an estimate. All that has to be done is obtaining p-value from the NULL model. I don't think there is a place for optimization (learning) here.
Of course the estimate itself is "learned" - like taking the mean of data. But it's not part of hypothesis testing itself. For example I can use "accuracy of a classifier" as an estimate and try to construct a p-value for it.
Your example
In your example (with real and placebo drugs) you construct a situation that has 2 classes therefore is applicable to classification. But this hides the hypothesis testing scenario. Better think of a hypothesis testing 101: toss a coin 10 times - what is the probability that the coin is fair? I don't see how this can be cast as a question for classification easily.
Or to change your example a bit - what if it was a paired experiment and the test is being done on the same people before and after treatment? It would not be correct to cast "before" group as one class and "after" group as another.
More problems
Also your statement:

If there is high enough correlation, I could determine that the change was significant.

Well - what is high enough? If I have mean difference of 0.5 and a standard deviation of 100 - there will be a lot of overlap. Yet with enough samples hypothesis testing will be able to detect this difference. But what to do in case of classification here? As an example - is 51% accuracy enough to say these are different?
Note
All of this is based on some reflection. Hopefully someone will provide a more formal or sourced answer.
