Compare and find the differences between DC23 and DC24 below, the better one is the one that suits your needs.
In this post, DC23 refers to DC23 Turbinehead, DC24 refers to DC24 Multi Floor.
Dyson DC23 vs DC24
images from manufacturer

Design
DC23 is a canister vacuum while DC24 is an upright vac.

Cord Winding
DC23 has automatic rewinding function while DC24 doesn’t.

Suction Power
DC23 has much more suction power.
DC23 has a suction power of 220 air watts while DC24 has only 115 air watts.

Motor Rating
DC23 has a 12 amp motor while DC24 has a 6.5 amp motor.

Brushbar Control Type
DC23’s brushbar switch is located on the head, thus you have to bend down to turn it on/off; whereas you don’t have to do this when using the DC24, since it
uses fingertip brush control.

Movement Type
DC23 uses fixed wheel while DC24 features Dyson’s Ball technology.

Tools
DC23 comes with a stair tool, crevice tool, and brush tool; whereas DC24 comes with a combination accessory tool.

Bin Capacity
DC23 has a bigger bin.
The bin capacity of DC23 is 0.44 gallons while DC24 has a 0.19 gallon bin.

Weight
DC24 is much lighter.
DC23 weighs 19.5 lbs while DC24 weighs only 11.65 lbs.

Cord Length
Both have short cord. However, the cord length of DC23 is a bit longer.
DC23 has a 21.6 ft cord whereas DC24 has a cord length of 20 feet.

Size
The dimensions of DC23 are 13.9″ x 19.3″ x 11.9″ while the dimensions of DC24 are 29.6″ x 11.0″ x 13.7″ (HxWxD).

Both of the two vacuum cleaners are priced at $399.99 at Dyson.com (prices checked on August 8, 2011), certified asthma and allergy friendly, come with lifetime washable HEPA filters, and carry 5 years warranty.

Overall, if you live in a small apartment, or want a light vacuum that is easy to carry, compact for storage, then DC24 can get the job done. Otherwise, go with DC23, because it has stronger suction and larger bin.