Well, the misunderstanding of how the peerage works is fairly fundamental, I'd have thought. I'm not quite sure if Sherlock is Lord Holmes or not in the early chapters, before his father dies, or if he only gets the courtesy title on becoming heir to the peerage, but even so for an Irish peer to have a seat in the Lords altogether means he's one of a minority even in a minority, so this sort of thing

Sherlock recognised, vaguely, a few names — Carson, Churchill, Asquith — though if he ever had known their positions, he has since scrubbed them from his mind, leaving behind not a single blot on the power of his memory

which is obviously intended to pick up on BBC Sherlock's "why should I care who the Prime Minister is?" falls away, since the most likely answer given the milieu into which she's put Sherlock is "Because you've got to meet him at the station, he's coming to stay."

BBC Sherlock ditches unnecessary knowledge; he's not just terminally ignorant for the sake of it.