Open episodes are used when you are telling an ongoing story. When you start watching an open episode, you have no idea or guarantee where it will end up. The Sopranos and Everwood are good examples of series that use open episodes.
Closed episodes follow a predictable formula. The end point of an episode is almost certain to maintain the status quo of a series. CSI and Law & Order provide good examples of closed episodes structure.
Now, there is a give and take between these two - mostly to do with series and season continuity and character growth. Lost, for instance, seems to have a closed structure when you think about its backstories but a reasonably open structure when you look at the overall story it's telling. It would be fascinating to see a Closed Series (such as a police procedural) with strong character growth... so that revisiting the same type of story gave no guarantee of the same outcome.
Of course there are advantages and disadvantages to both types of episode structures (which I'll expand on later).