There is no huge gap, where are you getting this from? The recent PS4 footage of the UI and performance clearly shows how fast it is and there's tons of new features and overall improvements. We will also see a full breakdown within the next 7 days from IGN of the PS4 UI/PSN so you will see more of what I"m talking about. These are completely different consoles, this isn't PS3 and 360.
I'm basing this mostly on previous gen, which I think will translate heavily into next gen. We aren't talking about new hardware built from the ground up. The networks will build mostly on what is already there. Xbox Live and PSN, while having different UIs on the new consoles, will be essentailly a cross gen service. I think Sony will close the gap a little this time but it will still be a significant gap.
I don't want to derail the thread since it is about Hardware performance and not a Live/PSN thread but I felt like I should defend my statement.