From what I can tell the isn't much difference. The 870 drops support for parallel ATA (old IDE ports), supports two more USB ports, and includes gigabit Ethernet. Both are connected with HyperTransport 3.0 but the 870 has A-Link Express III in the northbridge where the 770 has A-Link Express II.
Theoretically, SATA 6Gb/s has double the transfer rate from SATA 3Gb/s but real world results may vary and usually are never double.
Of course these are my opinions and someone may have a better point than I do.