There are a number of achievements that are fairly difficult to obtain. Having a record of these feats sitting right there on your Battle.net account is an easy way to show someone "Look! I did something pretty impressive!" But with everyone cheating, the value is gone. There is the personal satisfaction of knowing you obtained it legitimately, but if you show anyone your Battle.net account, they'll just think either you cheated or it's not a big deal since everyone else has the same "achievements."
It's the same thing as high scores in the days of yore. Like beating a game without losing a single life. Some people enjoy the challenge and sharing it with friends. Most game networks now have a way to verify that someone actually did something of noteworthy, whether it be an achievement or a trophy.
So yeah, they do nothing for gameplay mechanics. But they do a lot for why people play a game. To ensure ALL aspects of multiplayer, including achievements, are fair, Blizzard HAS to crack down on external cheats, even in single player.
Again, I'd like to reiterate that lawsuits may be a bit extreme. However, I do agree with their strong stance on the matter. People paid for a game and all its features. To have one feature ruined by others cheating and for Blizzard to nothing would be far worse than cracking down on people using external cheats.