Congratulations, first of all; a great prize. Nothing will make your mind up more than trying both out in the shop. I went to Waterstones only a month or so ago and did just that. I thought the 5" screen was too small and I'd prefer the 650 but (to me) the difference wasn't worth the price difference. I may also have been swayed unconsciously by the way the 650 demo model was faulty and wasn't behaving itself when I had a prod around on it!
The 5" screen is fine in use, surprisingly so. And very light and portable. I have mine in a (non Sony) leather case picked up off Amazon for about £15. A case is a must (the Sony is well built but I don't know how scratch resistant the screen would be).
But ultimately, you need to try both to see which suits you best.