I think in some ways video games take the mundane tasks out of a board game. For example, Monopoly on a computer rolls the dice, counts the moves, and performs transactions faster than a person can with the board game version. Because of this, you can get to the heart of the game much easier... For Monopoly, that would be what properties you purchase / trade for in order to give you an edge in acquiring a monopoly on a particular color group. etc. I think in part what bores people when playing board games is doing all of those mundanes tasks that a computer removes from gameplay; once you've mastered the art of rolling dice and moving your pieces, Chutes and Ladders isn't much fun anymore.
However I know some people like a more tactile interaction with the game, even if it is just rolling dice. Going back to Monopoly, with the board game if you are 8 spots away from landing on Boardwalk (and thereby completing your monopoly on those properties), it may be more fun to rub the dice, blow on them for luck, etc. Also with board games, they force you to play with other people... rarely will you find a board game that allows you to play by yourself.
Finally, although not strictly "board games" in the traditional sense, true roll-playing games that don't have too many rules offers the advantage of a true "sandbox" experience, where you can create the game you want as you're playing it. I remember when I was with a group of experienced gamers and a new game master putting that GM through a crash course of being able to think on the fly... He started off the story where we were inexplicably in a tavern listening to some old man ramble on about some caves that offered adventure and wealth. All of a sudden, our characters got bored with the "old geaser" and we decided we'd have more fun going outside of town and trying to catch a ship hunting down pirates. With a computer game, you usually can't be so dynamic in being able to do whatever you want. Even with some of the more so-called open-ended CRPGs, there still is usually a main plot-line, and all of the actions are controlled.