Confidence Interval is the interval in which the population mean is supposed to fall. Confidence Intervals are determined in all hypothesis tests as we infer something about the population from the sample.
Prediction Interval is the interval in which an individual value is supposed to fall. Prediction Intervals are determined when we use statistical tools for predictions.
Since Confidence Intervals are estimates for means, there are less chances of going wrong and hence it is smaller. On the contrary, since prediction intervals are point estimates, there are higher chances of going wrong and hence it is bigger than confidence interval.
Examples
1. Estimating the Sensex or Nifty level at the month end will be like determining Confidence Intervals, whereas, estimating the price of a particular stock at month end is like determining Prediction Intervals
2. Confidence Interval - estimating the overall sales for the product mix. Prediction Interval - estimating the sales for a specific product
Regression analysis when used for forecasting or predictions will yield both confident and prediction intervals. Usage of one over the other would depend on the output / variable that the organization is forecasting for. I would believe that most organizations work with confidence intervals, while prediction levels give them an indication of the best and the worst case scenarios.