When breaking down the economy, the first groups are sectors which describe a general economic activity. Then all of the companies that fall into that sector are categorized further into industries where they are grouped only with companies with which they share very similar business activities. This is not the end, however. Industries can be further sub-categorized into various, more specific groupings.
It should be noted that you may find situations in which these two terms are reversed. However, the general idea remains: one breaks the economy down into a few general segments while the other further categorizes those into more specific business activities. In the stock market, the generally accepted terminology cites a sector as a broad classification and an industry as a more specific one.