Zotac or gigabyte 560ti 448?

I would get the Gigabyte, because of there is warranties and BF3. But it would be better if you tell me exactly on what are you referring to.

I am not entirely sure what you are unclear about...

To try and clarify what I said, these are two non-reference nvidia gtx560ti 448 core graphics cards, and I'm going to use them to play games at 1920x1080 resolution.

Since they are both identical GPUs, I would say much a decision comes down to the cooler. Both how well it performs and the amount of noise it performs.
Not exactly sure which cooler Zotac uses though, from quick searches it should be the single fan cooler not unlike the reference one with orange mesh.
Gigabyte on the other hand uses their trademark Windforce coolers. There is a reason why Gigabyte sticks with their Windforce coolers through their higher end range, they are very good.
Hands down then cooling should go to Gigabyte.
OC wise, both will perform very well (as apparent from the Ti series) but, the Gigabyte unit will be less stressed when it comes down to temps again thanks to the Windforce cooler.
In my opinion, even with or without the BF3 voucher, I would go for the Gigabyte unit. A little peace of mind regardless when idling of OCed.  

OK, today I will make it extremely clear.

Show me the two card model you want to buy.

Uh, I may be wrong. But, don't Gigabyte and Zotac both only have one variant of the GTX560Ti Core 448?
Gigabyte's one with the Windforce 3 cooler and Zotac with their AMP! style cooler (oragne mesh on black shroud).

the gigabyte for 3 reasons: better cooler, warranty and BF3 which is an AWESOME game