The 870 is a much larger frame tractor then the 770.
A good comparison would be the Deere 790 (which is identical to the 770 except for the engine, tires and a few other odds and ends; and did replace the 770) and 990 (which replaced the 870/970 series tractors). Both the 790 and 990 are slightly larger and more powerful then the tractors they replaced.
The 770 has a 4 speed collar shift tranny with two ranges (8 gears fwd, 2 reverse), the 870 has a 3 speed trany with 3 ranges which comes to 9 speeds fwd, 3 reverse.
The 770 has 24 Engine HP with 20 at the PTO, the 870 has 28 engine HP with 25 at the PTO.
The 870 (bone stock) weighs about 600 lbs more then a bone stock 770.
The 870 has considerably more lifting capacity with both the loader and 3PH.
So, both nice machines and are very similar except for size and power. If in good shape, I'd go for the 870 in a heartbeat. But make sure it fits your needs. If you're buying to finish mow and maybe a little brush cutting, the 770 might be a better fit.
As an example, I had a 670 (770's little brother...same tractor with smaller tires and engine) which fit my 2.5 acres quite well. I wanted more power for a chipper and some field mowing I was doing, so I bought a 790.
Well, never bought the chipper and I'm not doing near as much field mowing as I did last year.
The better fit on my property was the 670.
If you have 5 acres or less and don't expect to use the loader for heavy work (real farmer type stuff), go for the 770. If more then 5 or 6 acres or you have a working farmette, you might prefer the 870.
So, a little more information about your intended uses would help, if you want advice on TBN.
As far as price...those MFWD 770's really hold their prices. But for $12500, you could buy a 790 (I did, for exactly that price). The 870 is maybe a little high..maybe not (depends on your location).
AKFish has a 970. I'm sure he'll share his experiences too.