Their conclusion specifically:
Soon, we will surely have a clear directive from the relevant medical bodies on this issue. In the meantime, our advice would be to play it safe — do not post such information online for all to see. Keep your stories, complaints, medically related commentary and your tongue-in-cheek dialogue for your personal friends.
So, the reason seems to mostly that they're afraid you'll be a victim of your own stupidity.
OK, we all know the scary stories about things gone horribly awry because of something written online: Jobs lost, divorces made nastier, friendships/family relationships strained. And in almost every case the blogger or poster in question was pretty stupid about it.
And you can be pretty stupid via a lot of different communications channels.
My advice? Be smart. Have a plan and a purpose and behave accordingly.
If your plan and purpose is just to vent? I'd consider an anonymous blog or a service that lets you make the blog private or only available to certain people you approve.
If your plan and purpose is to persuade and influence policy etc., then it's a totally different story.
Either way: Think really hard about what you're saying and if you care who finds it. I agree you can be found. Some stuff I am totally willing to put out there and live with the consequences...my political and ethical beliefs, no question. Some stuff is not worth it to me. Personal complaints and relationship issues? Not worth it.
Every person will draw their own boundaries.
Know yours before you start!