When done well, hyphenation draws little or no attention to itself; it helps to maintain consistency of line-length and improves the overall reading experience. When done poorly, hyphenation causes distraction, drawing attention away from the message and onto the typesetting. Today’s web browsers do hyphenation inconsistently and poorly (especially with full justification).
The practice of hyphenation goes back to illuminated manuscripts, hand-lettered on vellum (calf-skin), which was too costly to waste even the space of half a word at the end of a line.
With mechanically-set type, such as printed newspapers, hyphenation is a means of squeezing every last scrap of ad-space out of every narrow column-inch.
Modern page-layout software, such as Adobe InDesign, uses sophisticated algorithms to take multiple lines of text into consideration and makes decisions about word-spacing and tracking to use as few hyphenated words as possible without visibly distorting the text.
On the web, where pagination isn’t as important as in print, we may safely dispense with hyphenation in many situations. Web browsers (and their underlying operating systems) don’t (yet) offer the sophisticated approach we have at our disposal in print. At present, using it on body copy can work fine, while it’s best avoided in headings and narrower columns, depending on the context and layout.
Whether to hyphenate or not is a judgment call.