The difference between Roomba 650 and 620 are as follows.
Roomba 620 vs 650
images from manufacturer

On-board Scheduling
This is the key difference between them. The Roomba 650 has scheduling feature that allows you to set it to run automatically at a certain time for up to 7 times per week. This is convenient if you hate hearing the nosie. You can schedule it to vacuum when you are not at home.
While the 620 does not have this scheduler.

Accessory
The 650 comes with a Virtual Wall that keeps Roomba cleaning the areas where you want it to clean and stops it from entering off-limits areas.
While the 620 does not comes with the Virtual Wall.

Color
The 650 is in darker color while the 620 is lighter.

Otherwise, they are very similar.
They employs AeroVac cleaning system, Dirt Direct Series 1 to sense dirt, have a recessed one-hand carrying handle, etc.
And both of them are NOT compatible with the Room to Room navigation kit – Lighthouse, do NOT have full bin indicator.

Some differences between Roomba 880 and 620 that you may want to before buying the new 880.
Roomba 620 vs 880
images from manufacturer

Let’s look at some new things on the 880 compared to 620.

On-board Scheduling
The 880 has the on-board scheduling that the 620 does not feature. The scheduling allows you to preset the vacuuming time, then the Roomba will run at your desired time automatically.

Room to Room Navigation
The 880 supports the Lighthouse that helps navigate the Roomba room to room, this can make the Roomba vacuum multiple rooms more thoroughly and efficiently. While the 620 is not compatible with the Lighthouse technology.
If your house is small, then the Lighthouse function is not a big deal.

Bin and Filtration
The 880 is equipped with a larger dust bin to hold more dirt, and a HEPA filter to replace the standard filter on the 620 to trap more small particles and improve the air quality.

Dirt Detect Version
The 880 upgrades the Dirt Detect Series 1 on the 620 to Series 2.
The difference is that the Series 2 not only use acoustic sensor as the Series 1 has but also employs optical sensor to find dirt.
Furthermore, the 880 also features a Persistent Pass Cleaning Method that applies a back-and-forth cleaning pattern to clean where it finds excessive dirt.

Full Bin Indicator
The 880 also adds a full bin light to show you when it is full and needs emptying.

Home Base and Charging
The home base design is different now.
The one included within the 880 is an Integrated Home Base, it has no separate power supply, while the 620 comes with a Self-Charging Home Base with a Power Supply.

Battery
Both of them use the nickel–metal hydride battery.
The 880 however, upgrades the Advanced Power Battery on the 620 to a XLife Extended Life Battery that doubles the lifespan.

Included Accessories
The 620 only comes with a Flat Brush Cleaning Tool to remove hair and tangles from the brush while the 880 does not come with.
While the 880 comes with 2 Virtual Wall Lighthouses with C batteries included, 1 Extra HEPA Filter, and a Remote Control with batteries included.

Cleaning System
This is the key change between these 2 models.
If you own the 620, you can see that the 620 uses 1 Bristle Brush and 1 Rubber Brush in the bottom to pick up debris from the floor.
The 880 now does not use these brushes, instead it uses 2 Brushless Extractors.
And the 880 has the Airflow Accelerator design that creates a sealed channel that concentrates airflow to pick up more debris.
On top of that, 880 has an upgrade on the vacuum to generate more powerful suction.
iRobot claim that all the changes in the cleaning system will reduce maintenance and improve the cleaning results.

If you know more difference between Roomba 620 and 880, please share below.

Here are some differences between Roomba 880 and 595.
Roomba 595 vs 880
images from manufacturer

Carrying Handle
This is one of the apparent differences you can find from the appearance, the 880 has a flip-up carrying handle for easy one-handed transport whereas the 595 has no carrying handle.

Cleaning System
There is big change in the cleaning system.
The 595 uses counter rotating Bristle Brush and Rubber Brush to pick up dirt, dust, debris and pet hair from the floor.
Whereas the 880 uses Dual Brushless Extractors to grab and break down dirt and debris from the floor.
What’s the benefit of the new brushless extractors? Less hair tangle and less maintenance compared to the brush design.
The 880 also features a High Efficiency Vacuum and Airflow Accelerator.
iRobot claim that the new cleaning system provides greater air power and picks up more debris

Filtration
The 880 uses a HEPA filter to capture more tiny particles whereas the 595 uses standard AeroVac filter.
The filters are not interchangeable.

Dirt Detect Technology
The dirt detect system is different on these 2 models.
The 595 uses only acoustic sensor to sense dirt whereas the 880 adds the optical sensors to the dirt detect system, thus it has both optical and acoustic sensors to find dirt and debris.

Persistent Pass Cleaning Method
The 880 employs the Persistent Pass Cleaning Method that uses a back-and-forth cleaning pattern to vacuum spots where it senses more dirt.
While the 595 does not feature the Persistent Pass Cleaning.

Lighthouse Compatibility
The 595 is not compatible with lighthouse while the 880 supports Lighthouse.
What the Lighthouses do is to help Roomba navigate room to room.
Can the Roomba clean more than one room without the Lighthouses? Yes. But the cleaning will not be efficient, and the Roomba may miss some spots.

Full Bin Indicator
The 595 has no full bin indicator while the 880 has a full bin light to tell you when to empty it.

Battery
There is an upgrade on they battery.
The 595 uses the Advanced Power Battery whereas the 880 uses the battery which is called XLife Extended Life Battery.
What’s the difference? iRobot claim that the new battery can doubles the time before you may need to replace the battery.

Home Base
The 595 comes with a Self-Charging Home Base with Power Supply whereas the 880 comes with an Integrated Home Base for auto docking and charging.

Retail Price
The 595 retail price is much lower than the 880.
The 595 was sold at $299.99 at Costco, it has been discontinued.
Whereas the 880 is priced at $699.99 on irobot.com.

Included Accessories
The 880 comes with 2 Virtual Wall Lighthouses with C batteries included whereas the 595 comes with 2 Auto Virtual Walls with C batteries included.
The 880 comes with 1 Extra Filter whereas the 595 come with 3 Extra Filters.
The 880 comes with a Remote Control that the 595 does not include.
The 595 comes with 2 Brush Cleaning Tools that the 880 does not come with.
The 595 also comes with 2 Extra Side Brushes, 2 Extra Rubber Brushes, and 2 Extra Bristle Brushes while the 880 has no extra side brushes nor extra extractors.

If you know more difference between Roomba 595 and 880, please share below.