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.
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.
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. (...)
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.