The difference between Roomba 760 and 650 are as follows.
Hightlights: Cleaning System, Filtration, Accessories
Roomba 760 vs 650
images from manufacturer

Cleaning System
They both use the iRobot patented 3-stage cleaning process to vacuum, however, there is still difference on the cleaning system, they 650 employs AeroVac version 1 while the 760 employs version 2.
As part of the version 2, the 760 uses a redesign cleaning head that is claimed to provide better performance for picking up pet hair, dirt, allergens, etc. They said the cleaner head is designed particularly for pets.

Filtration
There is big change on the filtration.
Unlike the 650 which uses a standard filter – AeroVac Filter, the 760 uses Dual HEPA Filters inside the bin. The HEPA filters do better at filtering fine particles as tiny as 0.3 microns, thus improve the indoor air quality.

Bin
The 650 uses AeroVac Series 1 Bin whereas the 760 uses AeroVac Series 2 Bin. Not much difference on the capacity.

Price
There’s a $50 price difference.
The Roomba 650 costs $399.99 whereas the 760 is priced at $449.99 at store.irobot.com.

Extended-Life Power Management
The 760 features the Extended-Life Power Management software which is claimed to deliver up to 50% longer battery life.

Remote Control
The 760 comes with a Remote Control to let you control the Roomba with no need to touch it while the 650 has no remote control.
However, this kit is not that practical, there are a lot of users do not use it even they have it.

Accessories
The 650 comes with 1 auto Virtual Wall, 1 flat brush cleaning tool, 1 extra AeroVac filter.
The 760 comes with 2 Auto Virtual Walls, 1 flat brush cleaning tool and 1 round brush cleaning tool, 1 Extra Set of HEPA Filters.

Dimensions and Weight
The 650 weighs 7.9 lbs versus 8.4 lbs on the 760.
The 650 measures 13.39 inches in diameter versus 13.9 inches on the 760.

Color
The 760 is in lighter color.