basically run time and accessories.
the dc35 runs for about 13-15 mins on a full charge in regular mode and 5-6 mins in turbo mode.
the dc44 runs for about 18-20 mins on a full charge in regular mode and about 8 mins in turbo mode.
the dc44 also includes an extra accessory tool called the 'mini motorized tool', which works great for picking up pet hair and vacuuming upholstery.

The dc44 has stronger suction power, and lasts slightly longer.

I have both the dc35 and the dc44 and there seems to be two main differences: My dc 44 came with a motorized attachment for when I use it as a hand vac--the dc35 did not. Also the dc44 has considerably more power and suction than the dc35.

Has better suction

The dyson 44 has a longer battery life by about 5 minutes, than the 35. I think it also has more suction power but I don't know the exact details on that.

if you're asking if the charger itself stops working when the battery is completely charged, then i think the answer is yes and no.
the charging led status light goes off when the battery is fully charged, but i think the charger continues to top off the battery when needed.
i found this out by unplugging the dc44 after the battery fully charged and let it sit for a couple of days. when i used it, the battery died a lot faster than when i left it on the charger.

I put mine on a timer. Paper work says charge for 5 hrs. Make sure you let vac totally discharge.

Yes, shuts off automatically when fully charged.

YES

Yes, charger shuts down when charged.

about 20 mins on regular mode and about 8 mins on turbo mode

yes.. and no.
the dc44 does not have a quick release for its battery.
if you're looking to buy a spare, it wouldn't be a quick swap if you wanted to use one while the other was charging.
you'd also need to buy an additional charger for your 'spare' battery, as the dc44' battery only stays topped off while connected to its charger.
however, if your dc44's original battery finally dies, you can most likely easily replace it. a replacement battery for the dc44 probably runs around $69.

yes

Yes

Yes. Comes with a wall mount and charger

Yes the charger and a hanger are included. I do not use the hanger. My vacuum rests between my washer and dryer near a plug.

Yes it includes a charger.

Yes

yes

yes, 22.2v lithium ion battery.

Yes it does.

Everything detaches, you end up with the motor on hand. the blue wand detaches from the hand motor and the head (wide) detaches from the blue wand as well.

yes. the charger plug has an led status light. it's green when charging and goes out when it's fully charged.

The green light on the charger is on when the vacuum is charging. The light goes of when it's complete. Enjoy your vacuum!!

There is a light on the plug. When it is green it is charging when it is not lit it is ready to use.