The CSS property word-wrap (renamed to overflow-wrap in CSS3 to better reflect its intended purpose) specifies whether a word may be broken at an otherwise disallowed point in order to prevent overflow of the parent container.
The CSS property word-break specifies whether a break may occur within a word. I.e. between characters.
An example:
Original text
The text overflows the parent container because there are no word break opportunities on the line (space, hyphen, etc.).
break-word A word may be broken at an otherwise disallowed point in order to prevent overflow of the parent container.
break-word A word may be broken only if there are no acceptable break points on the current line.
Line 1. There is a valid breakpoint on this line (at the space character after the colon), so the long word (PI) will not be broken; it will be wrapped.
Line 2. There isn't a valid breakpoint on this line. The text will be broken mid-word.
break-all A word may be broken at any point, regardless of other break points on the line.