Why shouldn't I seperate my online persona(s) from my real life identity? What problem is Blizzard trying to solve here?
The Greater Internet Fuckwad Theory. People are less likely to act like an asshole if they don't have anonymity to hide behind. The only people who will be idiots any more are those who are idiots already.
Another example of a game using real names is iRacing, though real names are used throughout the game, instead of just on the forums. Again, the idea is to discourage anti-social behavior (intentional wrecking, profanity, etc), and this is part of their aggressive policing against griefers. In this case, such a prevention is especially necessary for a racing game of this type. I don't frequent the Blizzard forums, so I'm not sure if there are enough douchebags there to justify such measures.
I make it a point to avoid any forum identity that could be easliy traced back to my real name, because stuff comes back to haunt you. Do you really want somehting you said 20 years ago in some gaming forum to come up in a job interview?
In that case, either:
1) Don't be a jack-ass on the Blizzard forums.
2) Don't use the Blizzard forums.
That said, I've given my screen name which I've used for the last decade as part of a background check for an interview, so I'm well aware of the issue. However, nothing I said in the past (I've said a lot of stupid stuff) was an issue. If it is a concern to you, don't use the forums (it's an optional part of an optional game you play, you can deal with it), or simply police your behavior before it is recorded for all posterity on the Intertubes.