Losing the aesthetic appeal of the board game. There is something to be said for the appearance and literal feel of a board game. It engages you in a way no controller can. This problem is compounded by the fact that most board games are translated very awkwardly to consoles, neither going with a pure "video game" or a pure "board game" look, and usually winding up somewhere in the middle with what looks like a cheap mish-mash of the two. A game like Eye of Judgment does its best to actually add aesthetic value in the conversion (e.g., animations), the counterpoints being (1) who watches the animations more than once anyway?; (2) there isn't much of a tactile element to card games regardless (contrast to games with intricate pieces, like Warhammer). Usually there is a net loss here, if just for the tactile loss.
The social element. Obviously online play is great for many, as it opens up a world of players. You can play your friend in Idaho or the best player in the world without traveling to some dorky convention. Power of the Information Supherhighway. But beyond this clear benefit, the social aspects in general are completely lost online. To use a terrible analogy, playing Catan online is about as similar to playing Catan with friends around a table as playing Rock Band is to playing in a real rock band. No one has a mic. You can't see anyone. Or you can see people but it's through some Creepsville Central grainy webcam. You can't flip the board when you're about to lose. All the good stuff. Of course, there is also the simple issue that actual local multiplayer in games with hidden elements is often impossible or clunky, and leaves you wishing you just had the board.
Exploits, imbalances, and being "played out." What most people would view as a positive to video board games--the ability to cycle through play sessions 10x faster than tabletop sessions--is perhaps a negative to me. I can go on my PC and play 5+ sessions of Ticket to Ride in an hour, whereas the tabletop version takes me about an hour for a single session, and is something that's far more difficult to "get together." This is even truer for 3+ player games, where I can usually only play the tabletop when you have friends over. The issue I have with this is that it just whittles the game down to its barest play mechanics in no time. To be corny, the "magic" of discovering a game's strategies and imbalances over months or years of play sessions can be exhausted in one night at the computer. This is one of the big reasons I've stayed away from recent XBLA releases.
Winning the game is important when I sit down with friends or my wife, but to go back to the first point the social experience and everything that goes along with it is often more important. When you go online, it doesn't feel like you're diabolically plotting to ruin your another player's plans, or you're in whatever universe the game has created--it feels like you're playing math against each other.