To start with, let me describe my point of view.
What is Statistics?
Opinions vary. In fact, there is a continuous spectrum of attitudes toward statistics ranging from pure theoreticians, proving asymptotic efficiency and searching for most powerful tests, to wild practitioners, blindly reporting p-values and claiming statistical significance for scientifically insignificant results. Even among most prominent statisticians there is no consensus: some discuss the relative importance of the core goals of statistical inference, others comment of the differences between "mathematical" and "algorithmic" cultures of statistical modeling, yet others argue that mathematicians should not even teach statistics. The absence of a unified view on the subject led to different approaches and philosophies: there is frequentist and Bayesian statistics, parametric and nonparametric, mathematical, computational, industrial, applied, etc. To complicate the matter, machine learning, a modern subfield of computer science, is bringing more and more new tools and ideas for data analysis.On top of that, data science, a fancy mixture of statistics and machine learning is becoming more and more popular.
I tend to view statistics as a branch of mathematical engineering that studies ways of extracting reliable information from limited data for learning, prediction, and decision making in the presence of uncertainty. To the best of my knowledge, this view was first expressed by Cosma Shalizi.
Statistics is not mathematics per se because it is intimately related to real data. Mathematics is abstract, elegant, and can often be useful in applications; statistics is concrete, messy, and always useful. The difference between statistics and mathematics is akin to the difference between a real man and the Vitruvian. As a corollary, the proofs are not of paramount importance in statistics. Their main role is to provide intuition and rationale behind the corresponding methods. On the other hand, statistics is not simply a toolbox that contains answers for all data related questions. Almost always, as in solving engineering problems, statistical analysis of new data requires adjustment of existing tools or even developing completely new methods. For example, recent years witnessed an explosion of network data for which most of the classical statistical methods and models are simply inappropriate.