I've used blood from elsewhere, mostly when I couldn't resist picking a scab, but I was always told ideally it should come from the finger tips using a lance.
The people above have nailed most of the reasons but the other one I was told by my usual nurse was it's very easy to warm your hands up to get a blood flow if you're trying to do the test in less than ideal conditions, not so easy to warm up other areas. Don't do what Jonny did on his first ever solo outing for this, not making sure hands are warm (I now have a usb hand warmer I hold for a finger beforehand) , prick the finger, no blood, so I do it again, no blood, on go number three all three start bleeding at once.