Balisongs are banned because with a bit of practise, they're wicked fast. They're banned under the same law that bans automatic/flick/gravity knives.
AFAIK they're fine within your own home, but even having one about your person out and about is a chargeable offence, due to it being an offensive weapon regardless of context.
Tinkerton is spot on with the legal situation here in blighty though.