If you haven't resolved the issue, here's my solution (and I didn't have to resort to deleting Facebook or FB integration in iOS 6):
I had the exact problem- although the WiFi connection on my iPhone 5, tested via Speedtest app and through web browsing, has always been fast (@ 40Mbps), downloading media from iTunes and apps from the the App Store was abysmal- for example, a 250mb file took over 15 minutes to DL. After seeing a Genius at Apple Store, his conclusion was that it's probably my router since everywhere else (including at the Apple Store I was at) I had no problem downloading apps/media at much faster speed. He didn't exactly specify what could possibly be the problem with my router.
After going through the setting on my router, I discovered that at some point I had set the DNS to manual entry (instead of being set by the ISP) and was using the Google Public DNS which is/are 8.8.8.8/8.8.4.4. I had found this to be much faster and more reliable than the default DNS. But it occured to me that this could be the reason for the slow connection to Apple-specific network (iTunes/App Store). So basically I look for alternative DNS and entered this for my primarily 64.71.255.253 and for my secondary, 64.71.255.198.
And they work- I'm back to getting fast App Store/iTunes DL speed while the rest of my connection remained at the same speed and reliability as the Google DNS. My advice is to look into alternative DNS from what's currently on your router (especially if, like me, you've resorted to using Google DNS for speed/reliability). But if you're unfamiliar with setting your Router, I would either refrain or ask someone you know who has some basic knowledge in setting up router.