The main difference between them is the floor nozzle and the color.
The WS620 has a smaller floor nozzle than the WS632.
The WS620 is slate grey,
WS632 is steel grey,
WS632BLBRN is rose gold,
WS632GDBRN is gold,
WS632BLBRN is cove,
WS632PKBRN is thistle,
WS632GNBRN is sage.

images from sharkclean.com
Otherwise, they function the same.
These Shark Wandvac lightweight cordless vacuums function similarly to traditional stick & handheld 2-in-1 vacuums, but with a slimmer design.
They are designed for everyday small cleaning jobs.
You can use the hand unit which weighs 2.1 lbs only for quick pickup, or use the hand unit with the wand for longer reach, or connect the floor nozzle to the wand for cleaning hard floors and carpets.
- Have On/Off button which does not require you to keep holding down. But if you need extra power, you have to hold down the boost button.
- There is a normal power mode and a Boost button for pressing to get extra suction for tougher messes.
- They have Shark’s PowerFins brushroll which use silicone fins to replace traditional bristles to deliver continuous contact to floors and deep clean dirt, debris, pet hair & more.
- They also have self cleaning brushroll design which reduce hair wrapping issues.
If there is any blockage, you can remove the brushroll cover to access to the roller to clean the brushroll and floor nozzle.
- Brushroll cannot be turned off when vacuuming.
- No manual carpet pile adjustment control.
- They feature Shark’s HyperVelocity accelerated suction power for providing a good cleaning performance.
- They all come with one lithium ion battery which is detachable. If you want to extend the run time, you can buy an extra battery.
- One battery runs approximately 8 to 17 minutes. In regular mode, it runs approximately 17 minutes; with the boost button pressed, it runs approximately 8 minutes.
- Full charge takes approximately 3.5 hours.
- There’s battery indicator to tell you the battery power level when charging and in use.
- They use brushless motor like many other stick vacuums on the market.
- They are bagless, dust cup is small, holds 0.13 quarts, pretty small.
- The bin has one touch empty design.
- No HEPA filter. For those who are looking for one with HEPA filtration, this one is not your choice.
- No LED headlights. If you need one with headlights to help clean low light areas like under bed, please note that these vacuums do not have headlights.
- Though the floor nozzle is relatively big, the hand unit has the motor inside, so they are top heavy and do not self stand. When you need to take a break or stop to answer the phone or move something, you have to lean the vacuum against something or lay it flat.
- Wand is straight, length is not adjustable.
- Come with a Duster Crevice Tool to clean corners, along baseboard edges and other hard to reach tight spots;
and a Pet Multi-Tool which has stiff bristle brush to clean stuck-on debris, or you can remove the bristle brush to remove stubborn pet hair and large piles of debris from upholstery.
- Come with a charging base for charging and storage.
For this somewhat big footprint base, it is strange that it only holds one of the included attachments.
- You can charge the unit in an upright position or in the side by side position with the charging dock, or charge the hand vacuum without the charging dock.
- Carry 5-year limited warranty on unit and 2-year limited warranty on battery.
For more details, you can download the Shark Wandvac user guide:
https://m.sharkclean.com/include/pdf/manual-ws632.pdf