Applied statistics vs Mathematical statistics The Help Center for this site says we can ask question about, among other things, mathematical statistics.
I am curious to find out what mathematical statistics is. And I thought it might be easier for people to explain something in contrast to another thing. So the question I put forward is, 
What is mathematical statistics, as opposed to applied statistics?
 A: There are not only mathematical statistics and applied statistics, but also statistics (in general). You could say that statistics is about why and applied statistics is about how. Mathematical statistics is a branch of mathematics and generally a scientific discipline (the same as statistics). Applied statistics, on the other hand, is a term commonly used to name courses for non-mathematically oriented audience, that teach you how to apply statistical tools for the purpose of data analysis. You can find multiple applied statistics handbooks named like: "Discovering Statistics Using SPSS", "Statistics for Social Science" etc. Applied statistics is often applied by non-statisticians, e.g. researchers doing their projects. However, this doesn't mean that statisticians do not apply statistics, but rather it's applied statistics that is not interested in researching statistical theory, but rather it's applications. Statistics is concerned about statistical problems, while applied statistics about using statistics for solving other problems.
There are journals on applied statistics that promote development of statistical tools (see below).
Examples that could give you a scope on what applied statistics is:

Journal of Applied Statistics provides a forum for communication
  between both applied statisticians and users of applied statistical
  techniques across a wide range of disciplines. These areas include
  business, computing, economics, ecology, education, management,
  medicine, operational research and sociology, but papers from other
  areas are also considered. The editorial policy is to publish rigorous
  but clear and accessible papers on applied techniques. Purely
  theoretical papers are avoided but those on theoretical developments
  which clearly demonstrate significant applied potential are welcomed.
  The Journal aims for a balance of methodological innovation, thorough
  evaluation of existing techniques, case studies,speculative articles,
  book reviews and letters.

(source)
or:

The Journal of the Royal Statistical Society, Series C (Applied
  Statistics) (...) is concerned
  with papers which deal with novel solutions to real life statistical
  problems by adapting or developing methodology, or by demonstrating
  the proper application of new or existing statistical methods to them.
  (...) A deep understanding of statistical methodology is not necessary to
  appreciate the content. Although papers describing developments in
  statistical computing driven by practical examples are within its
  scope, the journal is not concerned with simply numerical
  illustrations or simulation studies. The emphasis of Series C is on
  case-studies of statistical analyses in practice.

(source)
or aims of applied statistics courses:

The MSc in Applied Statistics will aim to train you to solve
  real-world statistical problems. When completing the course you should
  be able to choose an appropriate statistical method to solve a given
  problem of data analysis and communicate your results clearly and
  succinctly. The course aims to equip you with the computational skills
  to carry through the analysis and answer the problem as presented. (...)

(source)
I didn't give here a broad review on what statistics or mathematical statistics are, but it should be self-explanatory since I given you examples on how does applied statistics differ from them.
A: If you're simply taking a book on techniques and learning how to plug things into R/SAS/STATA, you're really not doing applied statistics, imo. The best applied statistics courses/programs involve a healthy dose of theory. The difference between the two really is one of emphasis. In an applied course, you're learning to use techniques, whereas in "pure" stats courses you're learning to develop or prove things.
So in a good applied regression class, you'll see the proof that, say, the coefficients in OLS regression are the MLE, assuming the errors are Gaussian. You might remember this, along with the implication that this means the coefficients are efficient estimators, as reasons that OLS is "good." But aside from remembering this (maybe you get a question on the test worth a few points as to why OLS is "good," and you'll have to write down this fact), and maybe a homework question asking you to prove some implication of this fact, you're unlikely to use that proof again.  
In a "pure" stats class, however, the proof is the class.  You might not even see how this fact is used in the "real world." You're more interested in how the proof works, because at some point in the future you might want to prove that the estimator you're developing is an MLE.
Put differently, at my university, the graduate Applied Regression course uses a fair amount of proofs in class, and you have a hard time passing if you don't know calculus and linear algebra.  But the homeworks and exams are mostly about interpreting data.  In the PhD-level GLM course, however, they never so much as download a dataset.
