It provides a clear way to delete text for users who are not accustomed to keyboard shortcuts to select all text in an input field. Additionally, assuming the value of the search input is not erased after a search has been carried out, it can potentially provide the fastest way to perform a new search for the non-power user demographic.
For example, on Firefox, when you return to the search box, it automatically selects the previously searched input meaning I can start typing the new search input immediately. However, for someone such as my mother who will reluctantly step through an operation on a computer, this may result in her first pressing Delete/Backspace before typing their new search as they perceive that to be a safer and more predictable route towards achieving their goal.
Alternatively, if a clear button was available, by being pressed first it deletes the existing search input and causes focus to be retained for the input resulting in the search being carried immediately in two steps as opposed to three.
Best case scenario for each method is the second search takes the same amount of steps to be carried out, but there does exist a possibility by not providing a clear action that a redundant step may be carried out by less experienced/confident users.