Authentication issues with WiFi is usually because the wireless key, whilst you are sure you entered it correctly, actually does not exactly match in terms of uppercase or lowercase.
Could be when typing the wireless key on the iPhone you are typing an uppercase letter instead of a lowercase (or vice versa), as the iPhone 5 on-screen keyboard doesn't actually change the appearance of letters (they always appear as capitals, all you have is an indicator to show which case you are in).
The Android keyboard on the other hand clearly shows you an uppercase or lowercase keyboard.
So try typing the wireless key again but make sure the uppercase/lowercase indicator is set correctly for each letter before you type it.