This post provides a good explanation about the Prediction interval and confidence interval. I just forwarded it here to keep a record. The original author has many other enlightening posts about forecasting as well, which are interesting to check.
One example I can think of is in simple linear regression. For example, if I give you the value of the covariate for a future point, and then we would like to predict the value of the response. The confidence interval for this response is used to cover the expected value of y. But the prediction interval is used to cover a random future value of y. As a result, the prediction interval is wider than the confidence interval.