Big data and data mining are two different things. Both of them relate to the use of large data sets to handle the collection or reporting of data that serves businesses or other recipients. However, the two terms are used for two different elements of this kind of operation.
Big data is a term for a large data set. Big data sets are those that outgrow the simple kind of database and data handling architectures that were used in earlier times, when big data was more expensive and less feasible. For example, sets of data that are too large to be easily handled in a Microsoft Excel spreadsheet could be referred to as big data sets.
Data mining refers to the activity of going through big data sets to look for relevant or pertinent information. This type of activity is really a good example of the old axiom "looking for a needle in a haystack." The idea is that businesses collect massive sets of data that may be homogeneous or automatically collected. Decision-makers need access to smaller, more specific pieces of data from those large sets. They use data mining to uncover the pieces of information that will inform leadership and help chart the course for a business.
Data mining can involve the use of different kinds of software packages such as analytics tools. It can be automated, or it can be largely labor-intensive, where individual workers send specific queries for information to an archive or database. Generally, data mining refers to operations that involve relatively sophisticated search operations that return targeted and specific results. For example, a data mining tool may look through dozens of years of accounting information to find a specific column of expenses or accounts receivable for a specific operating year.
In short, big data is the asset and data mining is the "handler" of that is used to provide beneficial results.