Practical vs Statistical significance In a nutshell, what would be the main differences between practical and statistical significance? What would be the relative importance of each in drawing scientific inferences, for example?
 A: In a nutshell:
Suppose you are testing the effect of drinking a glass of wine per day on the risk of a heart attack. In layman terms, statistical significance refers to whether drinking affects such risk. Practical significance refers to whether such change in risk is of a relevant magnitude for real life concerns. If this is not the case, then you should not be worried about drinking one glass of wine per day, as the effect it has on your health is too small to be a concern.

More precisely, say you find, for a given significance level, that the effect of drinking is non-zero (e.g. p-value below 0.05). Then, the effect of wine on risk of heart attack is statistically significant.
Say however, that such effect, albeit non-negative is "very small". For example, say that drinking one glass of wine per day increases your risk of heart attack by 0.01%. Then, such effect has no practical significance. Conversely, if such effect were to be "large" (e.g. a 20% increase), the effect has practical significance. (the qualification of "very small" and "large" is usually estimated based on the standard deviation of the outcome variable in the sample/population)
In economics, the term for the latter is "economic significance". It is common to ask, after finding a statistically significant effect, whether such effect is economically significant. This is sometimes seen by exploring how much one standard deviation change in the regressor/control relates to the distribution of the dependent/outcome variable.
A: This is a large topic but I suggest that the following are key points.
The relation between practical and statistical significance is not well described in terms of relative importance.  To think in those terms rather suggests that a researcher has collected some data, used the data to test a hypothesis chosen without too much thought (perhaps that a regression parameter differs from zero), found that the test points to rejection of the hypothesis, and then asked whether this finding is significant.  Having reached that position it is certainly appropriate to assess practical significance, rather than assuming that statistical significance implies practical significance.
However, a better approach in the context of scientific research is to consider practical significance much earlier, at the stage of research design.  A hypothesis can then be chosen such that whether or not it is found to be correct is of practical significance. In the case of a regression parameter $\beta$, for example, this might lead to the choice of a null hypothesis as $H_0: \beta > \delta$, for some value $\delta$ judged to be a threshold of practical significance, rather than simply $H_0: \beta = 0$.  Of course the appropriate value of $\delta$ will depend as rocinante says on the field of study, and also on the particular question within that field.  Once a hypothesis is selected, consideration can then be given to data collection, and having collected data the hypothesis can be tested, the result being assessed in terms of statistical significance.  
A: Some points about the practical importance (I'm trying to avoid significant here) of statistically significant findings that come to my mind:


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*With increasing sample size, small effects will become statistically significant. Whether they are practically of importance won't change.
This is another way of looking at @Adam Bailey's point.

*On top of that, I see a second level of practical significance:
Roughly speaking, "statistically significant" means it is unlikely to observe such (or more extreme) differences if the null hypothesis is true. This is also the case for the carefully formulated Null hypothesis that takes into account the effect size that would be considered as practically important according to the previous point.
However, usually this is not all that I as your reader am interested in. What I want to know is not only what your findings are but also how much I can rely on your conclusions. That would be more like the predictive value of your results. In order to judge this, I'd need to know the "prevalence" of true hypotheses among the hypotheses you generate and test.
Which is a way of saying that testing a huge number of more-or-less randomly generated hypotheses usually is not going to help much, and why the total number of tests conducted is of huge importance for the practical conclusions you can draw from a statistically significant result. 
A: Statistical significance always becomes more likely the larger the sample size is. This holds since almost no experiment is done exactly under the null hypothesis. Small irrelevant differences can never be ruled out. Consistency of the tests can make such irrelevant differences significant. 
Practical relevance depends on the scientific fact, not on statistical convergences. In applied sciences, effect sizes are irrelevant if they cause no difference in the expert's judgement.
But both concepts can be brought together: Statistical tests for relevance can often be constructed by choosing an interval of irrelevant effect sizes.  Significant relevance at $\alpha$ level can be concluded, iff a $1-\alpha$-confidence interval is disjoint from this irrelevance interval. This way, enlarging the sample size increases the chance of statistical significance iff the effect is relevant. Otherwise even a confidence interval of length $0$ stays inside the interval of irrelevance.
A: Nutshell is being studied in relation to, among other things, prevention of diabetes and obesity. 
Say you wish to analyse the difference between rats that get nutshell extract in their feed and rats that do not.


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*Then the 'effect of the nutshell' is statistically significant if the results are unlikely given the hypothesis that there is no effect of the nutshell.

*And the 'effect of the nutshell' is practically significant if the results show a large size difference (estimated), 'large' being defined in relation to practical purposes.


Importance in drawing scientific inference: Statistical and practical significance do not need to occur at the same time (depending on the test size and variation of the population). It depends much on the estimated standard error for the effect size. If the SE is small then very tiny (practically insignificant) effects can still be tested as statistically significant. And if the SE is big then large (practically significant) effects can be tested as statistically insignificant (not because the effect is small but because the test is a bad detector). 
So errors are being made if these different types of significance are mixed: 


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*Arguing that no effect is present because of low statistical significance, while the test might have been not accurate enough to detect a practical significant difference.

*Arguing that a (practical )significant effect is present because of high significance, while the test might have been extremely sensitive to detect tiny effect sizes with little practical significance.

