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DC attaches straight to your PSU.
AC runs independent of the PSU, unless you run a relay.
So while the DC pump would only run while your computer is off, the AC would keep pumping regardless of whether the PC is on or not. Assuming the DC pump is connected straight to the PSU.
A good DC pump (such as a DDC or a D5) usually costs over $75 USD brand new.
A good AC pump (such as a 1048) usually cost lower than that (about $50 for the 1048)
From noise recordings alone, I'd go with an AC pump (specifically 1048 or 1046) because I've never heard a DC pump recording that had a noise pitch as low (and unobtrusive) as a 1046. 1048 has a slightly higher pitch.
If you want a DC version of the Eheims, Watercool, AquaComputer and Innovatek make them, but they all cost more than $100 USD.
DC...
A properly isolated Laing DDC is near silent. I am running w/o case at the moment and i cannot hear mine at all.
no relays, no second power cord, less footprint, whats not to like? A little more power draw is no big deal. you should have a good PSU already if you are thinking about watercooling (along with a decent system) or you are just putting the cart before the horse.
DC pumps create electical interference on the PSU rail they are attached to. It is very slight though. AC pumps usually cause EMI interference, so place them away from HDD's and other sensitive devices.