The 'heavy' balls exceeded the USGA's weight limit for golf balls (45.93 grams) by a fraction of a gram, or approximately half the weight of a U.S. dollar bill. In theory, the heavier ball will travel farther if struck solidly, but not enough to make a noticeable difference.
Unless you're competing in professional or high level competitions, you can continue to play the 'heavy' balls with the old sidestamp. It still conforms to the Rules of Golf and can be played in handicap rounds. Only those golfers playing in competitions that adopt a specific Condition of Competition must use the ball with the new stamp, beginning in June.