The iRobot Roomba 880 is latest Roomba robotic vacuum model.
It features the new AeroForce Performance Cleaning System which use 3 technologies to improve the cleaning:
Tangle-Free AeroForce Extractors: these 2 rubbery Extractors that are brushless replace the bristle brush and flexible beater brush on 770. iRobot claim that these Extractors pick up more from floors and require almost no maintenance because they eliminate tangled hair.
Airflow Accelerator: it helps concentrate airflow between the Extractors to draw in more debris.
High-Efficiency Vacuum: iRobot claim that this new vacuum on Roomba 880 provides a 5X improvement in air power than the older Roomba models including the 770.
Roomba 880 vs 770

Price:
iRobot Roomba 880 list price on irobot.com is $699.99 while the price of Roomba 770 is $499.99. (prices on 02-14-2014)

Battery:
iRobot Roomba 880 use the XLife Battery that claim to optimize the cleaning efficiency, result in delivering twice as many cleaning cycles as Roomba 770 over the life of the robot.

Bin:
iRobot Roomba 880 features a larger bin which can hold more dirt than the Roomba 770.

Lighthouses:
iRobot Roomba 880 support the Lighthouse technology that allows Roomba to vacuum from room to room while the 770 does not have this ability.

What’s in the box (when you purchase the robots from irobot.com):
iRobot Roomba 880:
1 iRobot Roomba 880
1 iRobot XLife Battery
2 Virtual Wall Lighthouses (batteries included)
1 Integrated Home Base
1 Remote Control (batteries included)
1 Extra HEPA Filter

iRobot Roomba 770:
1 iRobot Roomba 770
1 Rechargeable Battery
1 Battery Charger (3-hour charge time)
1 Compact Self-Charging Home Base
2 Auto Virtual Walls (each requires 2 C batteries, not included)
1 Extra Filter
2 Brush Cleaning Tools
1 Remote Control
1 Instructional DVD

Similar features:

• Fiter: HEPA filter;
• Dimensions: 13.9 inches in diameter, 3.6 inches in height;
• Weight: 8.4 pounds;
• Warranty: 1-Year Manufacturer’s Limited Warranty On Robot, 6-Month Manufacturer’s Limited Warranty On Battery;
• Dirt Detect Series 2 with Persistent Pass cleaning method;
• Control Panel: button control;
• iAdapt Responsive Navigation Technology;
• Self-Charging;
• Cleans On Schedule;
• Edge-Cleaning;
• Full Bin Indicator to tell you when to empty the dust bin;
• Spot Clean Mode to clean a small area quickly;
• Power Management to deliver longer battery life

If you know other difference between iRobot Roomba 770 and 880, please share with others below.

The major difference between the iRobot Roomba 880 and 780 robotic vacuum is the cleaning system.

According to the info from irobot.com, the Roomba 880 cleans floor up to 50% better than the previous Roomba models due to its AeroForce Performance Cleaning System that use Tangle-Free AeroForce Extractors, Airflow Accelerator, and High-Efficiency Vacuum which produces greater air power than previous models.

The Roomba 880 feature Tangle-Free AeroForce Extractors – dual, counter-rotating, rubbery, brushless.
The brushless design prevents the tangles and jams of hair, makes maintenance as hassle free as possible.
You can see the extractors below:

Whereas the Roomba 780 use 1 bristle brush and 1 flexible beater brush to pick up pet hair, debris, allergens, dirt and more.
Roomba 780 brushes:

Battery:
The Roomba 880 use iRobot XLife Battery that claims to deliver twice as many cleaning cycles as Roomba 780 battery.

iRobot Roomba 880 vs 780

If you know other difference between iRobot Roomba 780 and 880, please share below.