One of the obvious cases is the one mentioned by Nirav Chadda's answer, which can be shortened to "1 click is better than multiple key strokes". This is related to less effort and better usability
Another case can be seen on Yahoo, in this specific case you can see Yahoo's Basketball Fantasy page.
For this purpose, let's assume I'm an user who wants to add a new player to the team. So I'll click a button labeled "Add Player" and then I'll be directed to the following screen:
This screen displays all players based on some default filters and sorting, in this case, All Available Players sort by Pre-Season rank.
However, let's say I want to look for some player which I don't remember if named Devin or Devon, so I'll look for Dev , type part of the name and this is what I'll get:
Now, I have a list of 4 players. Please note the search field retains my "dev" search and added a "delete" icon (arguable but for me this is a close icon). I check the players and they're not a good fit for my team, so I want to get back to the list of ALL available players. So I click the icon and the first page shows up again (or if I used a different filter, whichever filter I have used).
So, this clear/close/delete icon not only deletes the characters in the search field, but also clears all dynamic info created on the search process (eg, AJAX'd processes).
An icon clearing only the inputted string would leave me with all the results I searched for. But... if I clear that field, it's safe to assume I don't want its results either. This could be easily solved by adding another button to go back or using the back button in user's browser. However, the clear icon simplifies everything in an elegant way providing great usability and less effort for user