Windows Phone 8.1 Update 1 has been released to developers, and brings Cortana to countries outside of the United States such as the UK and China. Being from the United Kingdom, I can finally try out Cortana, and from what it looks like, she doesn't appear to have a personality.
Cortana is known for being witty, comedic, and awesome. She'll answer you when you ask her who her daddy is, or what her name is. Cortana in the UK doesn't appear to do this at all, asking her questions like "who's your daddy" will result in a Bing search, same goes for asking her to tell me my name, or anything that requires her input.
She still does tasks, sets reminders, and such, so the 'personal assistant' side of her is still around, but the thing that differentiates Cortana from the competition is her own unique personality. This is something users outside of the United States don't appear to have.
I asked Twitter, and I'm not alone in this either. Many people are reporting that she doesn't respond to anything but tasks. Hopefully Microsoft just needs to flip a switch to enable her personality. If Microsoft have done this on purpose, then I'm not sure how I feel about Cortana as a whole. But clearly Microsoft is still working on improving Cortana -- especially outside the United States.