Please refer to this paper "Data Mining and Statistics: What's the Connection?" at http://statweb.stanford.edu/~jhf/ftp/dm-stat.pdf

Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. (refer to the Wikipedia article on Statistics.)

While interpreting and  analysing large sets of data, for prediction and pattern recognition purposes, Data Mining is used. Data Mining is just a computational method for statistical analysis.

Having said that, the second part of your question is quite redundant, since Data Mining and Stastics are conceptually different from each other and cannot be compared to each other.

the aim of data mining is to extract knowledge from statistics results

Statistics in the 1800's and early 1900's actually dealt with large data sets. All the theory was geared towards large (census) data sets. That's what made t-tests so different. 

Data mining is a combination of computer science (to access the data) and statistics (to analyze the data). 

Data Mining refers to the generating some set of data from which formulation or question is formed. It is similar to looking for something but exactly dot know. Or in other words data mining is a reverse process of statistical analysis.  In statistical data analysis, formulation is applied on the set of data to get the result. The formula refers exactly what we are looking for; whereas Data Mining refers to generating sub-set of data on which formulation is fit to refer what exactly it means. 

Please refer to this paper "Data Mining and Statistics: What's the Connection?" at http://statweb.stanford.edu/~jhf/ftp/dm-stat.pdf

Data mining is carried out by a person, in a specific situation, on a particular data set, with a goal in mind. Typically, this person wants to leverage the power of the various pattern recognition techniques that have been developed in machine learning. Quite often, the data set is massive, complicated, and/or may have special problems (such as there are more variables than observations). Usually, the goal is either to discover / generate some preliminary insights in an area where there really was little knowledge beforehand, or to be able to predict future observations accurately. Moreover, data mining procedures could be either 'unsupervised' (we don't know the answer--discovery) or 'supervised' (we know the answer--prediction). Common data mining techniques would include cluster analyses, classification and regression trees, and neural networks. 

Statistics is concerned with probabilistic models, specifically inference on these models using data. Statistics is just about the numbers, and quantifying the data. This is normally carried out by machines. There are many tools for finding relevant properties of the data but this is pretty close to pure mathematics.

There is no clear cut and absolute difference. It is all about different communities of interest. Some of the best work in statistics is associated with data mining, Hastie, Tibshirani and colleagues from Stanford are world leading statisticians actively pushing forward the field of data mining.

There are both statisticians and computer scientists who describe themselves as data scientists. If there is a difference - and this is a generalisation which is by no means universally true, those involved in data science from a statistical background tend to be more interested in rigour; those from a CS background tend to be more interested in practicality - their focus is more on engineering.   

Data science is not just statistics (or re inventing well known results in statistics). It is also concerned with the computational processes and algorithms required to manage and process petabyte scale data. Though certainly many statisticians are actively involved with petabyte scale data.

Of course, what we need is a combination of rigour and practicality. The most important thing is to use rigorous methods to provide practical and reliable solutions to real problems. Who cares if we describe ourselves as statisticians or data scientists in doing that?