The advantage of getting a warranty replacement from Apple over Verizon is that, if you make a Genius Bar appointment and Apple determines that your phone is covered by warranty, they will replace the phone on the spot. Verizon will have to order and ship a phone to you. In both cases, you will have the remainder of your original one year warranty or 90 days, whichever is longer. If you have purchased AppleCare or purchased an extended warranty through Verizon, you will fall under the terms of that product.
I have no idea if the phones are remanufactured at the same place. I've had certified pre-owned iPhones from Verizon and a replacement iPad from Apple. Both worked perfectly.