I can confirm that a bad backup can fubar your phone. When I got my iPhone 5, I couldn't send regular text messages- only iMessage to other idevices. There was a corrupt setting I. My iPhone 4 backup. I had to dump and start from scratch.
Here is something else to consider. You need to put both devices side by side on a table and test at the same time. There is going to be slight variances, so you need to run a series of tests with speediest.net. They should average about the same. If they do not, then you need to go to another wifi source and do it again. If there is still a difference, call Apple support and tell them your problem.
Now, if the majority of your problems seem to stem from Facebook -accessing videos and websites, then it could be the app. Delete it, sync your phone and reboot. Then add it again and see if that makes a difference.
FB on my phone crashes at least once a day and its either lightning fast or slow as hell.
That brings another point. Make sure that your as close to comparing Apples to Apples when doing tests. If the Facebook on the iPad hasn't been updated at least once since last summer, then you have different versions and that could be the problem too.
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