The difference between Roomba 780 and 870 are as follows.
Major Ones: Control Pannel, Cleaning System, Filtration, Bin, Lighthouse Compatibility.

Roomba 870 vs 780
images from manufacturer

Control Panel
One of the difference between these iRobot vacuum cleaners is the control panel, the 780 uses Touchpad Control whereas the 870 employs Button Control.

Cleaning System
They use different cleaning system.
The 780 uses AeroVac Series 2 Cleaning System whereas the 870 uses the AeroForce Performance Cleaning System.
According to iRobot, the new system creates more powerful suction and the brushless design reduces the maintenance to minimum.

Filtration
The 780 uses 2 Small HEPA Air Filters versus Single Large HEPA filter on the 870.

Bin
The 780 uses the AeroVac Series 2 Bin whereas the 870 uses the sealed, HEPA-filtered bin that is larger and easier to clean.

Lighthouse Compatibility
The 780 supports the Virtual Wall Lighthouse that help navigate the Roomba to clean from room to room whereas the 870 does not support the Lighthouse.

Charging
The 780 uses a Battery Charger and a Compact Self-Charging Home Base whereas the 870 uses the Integrated Home Base.

Battery
The 780 uses the Advanced Power System Battery whereas the 870 uses the XLife Extended Life Battery that was claimed to have longer lifespan.
The Xlife battery is $10 more expensive than the APS battery, and it is compatible with the 780.

Accessories
The 780 comes with a Remote Control that the 870 does not include.
The 780 comes with 2 Virtual Wall Lighthouses whereas the 870 comes with 2 Virtual Walls.
The 780 comes with 2 brush cleaning tools (Round and Flat) to clean the bristle brush whereas the 870 does not include.
The 780 comes with 1 Extra Bristle Brush, 1 Flexible Brush and 1 Side Brush whereas the 870 does not comes with any extra extractors or side brushes.
The 780 comes with 1 extra HEPA filter set (2 filters) where the 870 comes with 1 Extra HEPA filter.

Similarities:
priced at $599.99 at irobot.com,
features Full Bin Indicator to tell you when to empty the bin,
have scheduling function,
auto docking and recharging,
Dirt Detect Series 2 that uses acoustic and optical sensors to find dirt,
Persistent Pass Cleaning Method to clean dirtier spots with more time,
weighs 8.4 lbs,
13.9 inches in diameter, 3.6 inches in height,

If you know other difference between Roomba 870 or 780, please share below.