While it is extremely impressive that one could do so(given that such power for such price was unheard of until very recently), it seems rather pointless to compare these two boards on the basis that you are planning on plugging in enough peripherals to use them as your next desktop. Aside from the plummeting cost and wide availability of ludicrously overqualified x86s; the most stripped down ARM SBC is obviously going to require costlier and uglier peripheral tangles than the less stripped down ones(the CHIP has no 'sibling' designs; but is similar to Allwinner-based boards from others; the rPi zero has otherwise-identical models that add the USB hub and ethernet for you, or slightly punchier options).
The only reason to go with either of these is because you have some more constrained purpose for which the low cost and small size make the difference. So, do you want HDMI support; at the cost of bringing your own NIC; or do you want the NIC; but HDMI as an optional extra? Depends on whether you are building a headless project or not.
The other matter is the software support; which is as yet an unknown. The rPi has some Broadcom blobs that are very, very, unlikely to go away; but benefits from a known, solid, supporting ecosystem. The Allwinner A8 is a bit of an unknown quantity: the project claims to be aggressively mainlining everything(which would be extremely attractive); but Allwinner's GPL compliance has been...haphazard...at best in the past; and the MALI-400 is ARM's toy, not theirs, so they have limited control over that. If the CHIP's aspirations bear out, then it will have the distinct advantage of working with mainline kernel and u-boot. If they don't, or do only in part, then the question becomes one of 'which slightly oddball BSP is better?'