I am having the exact same issues. I have an iPhone 5s and an iPad 4. Both were connecting fine to the wi-fi in my house until two days ago. I updated (ie, was force fed) both devices to iOS 7.1.2 three, maybe four days ago. Now, I either cannot connect to my home wi-fi or if I can, the connection is dropped within minutes. My wife's Samsung galaxy can connect (and stay connected) anywhere in the house, including upstairs which is 15 meters and two walls away from the router.
This is bulsht. I've gone through Apple's stupid checklist, including start/restart, hard starts, and a "Reset Network Settings" which of course erases all my wi-fi locations and passwords that I've accumulated up to now.
This problem started with the iOS 7.1.2 update, clear and simple. I f**king hate this. I cannot connect, my kids cannot connect, and I can only get work done sitting down in front of my computer. Hey Apple, listen up. I am NOT an Apple fanboy/sheep who is locked into your ecosystem. Fix this problem NOW or I'm done with your products. I don't want to use Android but will switch in a heartbeat if you people don't make this right.
You folks KNOW this is related to the update, no matter what your "genius'" say. Fix this or I'm gone (yeah, that means no purchase of the iPad Air or the iPhone 6, both of which I was contemplating but now?)!