Data mining is a somewhat misleading term. The goal of data mining is to look for patterns in the data, not the data itself. This is done with methods from statistics, computer science and Machine Learning. Like BI, data mining uses descriptive and diagnostic analysis, but it also draws on predictive analysis methods.
The concept of data mining emerged sometime in the 1990s and grew out of the idea of searching for useful information in companies' (relational) databases. Today, the term is less used and has been replaced by the term "data science". One could debate whether there is a difference between the two, but broadly speaking they are similar.