The 5850 does not need much voltage for great overclocks. Mine is one of the worst samples but does 880Mhz @ stock volts and 950Mhz @ 1.175V which is 6870's stock voltage and it does not get loud at all. Only when pushing for 1ghz I need to increase the fan manually. Its almost a year ago when I had both cards. RE5 benchmark is quite a bit faster on the 5850 at 950Mhz compared to 6870 at 1000Mhz.
Both are good cards. 5850 is stronger for shader intensive games like BF3, the 6870 will fare better for tessellation intensive games like Crysis 2.
how much voltage you need is entirely luck of the draw, one person may hit 5870 clocks with stock voltage, another person might need more voltage just to hit 750. And again, whether or not you even can overvolt depends on brand and model.
Also, if you have a reference 5850 with the reference cooler still on it, then you are straight up lying about it not getting noisy, or else you live over a club or something and/or use UltraKazes or Deltas to cool your other components.
if you have a rare aftermarket 5850 with cooler that actually allows for reasonable decibels and temps under load,
and the model still allows you to overvolt, then you are lucky. I had to mod my reference 5850 to get proper cooling and noise under control. Granted, once I did so I could easily break 1GHz.
Also, are you sure about BF3? the only head to head comparison I can find shows the 6870 with a very slight lead at all resolutions, again, after overclocking I'm sure the 5850 would pull a lead, but I doubt it would be by anything earth shattering, and again we run into the issue of whether or not substantial overclocking is possible, so without knowing that we can only say they're even at best, if no a slight advantage in favor of the 6870.
Also, RE5 should be easily maxed with far more than 60fps average by either GPU as long as the CPU isn't holding things back (of which either GPU would feel identical then), and I doubt the 5850 is substantially faster, last I saw (RE5 isn't exactly used much anymore for GPU stress testing for obvious reasons) the 6870 could actually keep pretty close to the 6950 or 5870 (ie within 10% while maintaining an average framerate over 120 @ 1080p).
Don't get me wrong, I still have and love my 5850 for being my first and still only GPU to break 1GHz, and the fact that its 2+ years old and still relevant is also extremely awesome. However I had to do more work than what would worth it to be comfortable overvolting and overclocking it under these current circumstances. Back then I spent only a small fraction of the MSRP on an Accelero S1 and 120mm fan, today that same mod would run nearly half the cost of this upgrade scenario.
If the OP's option includes a brand/model 5850 that would be conducive to heavy overclocking with overvolting, and he'd be willing to do so, then my recommendation would likely swing in favor of the 5850. However we also have to keep in mind that power consumption and temps are only similar when both cards are at stock, once we start to crank them up the 5850 will
quickly outpace the 6870 in that regard.