The Dyson DC59 is better. It uses the new 2 Tier Radial cyclones technology and equips with better motor that makes it generate more powerful suction; improved battery to provides longer battery life; some parts of the unit are redesigned to make it more convenient to use such as the filter design and Trigger design; improved floor tool is more stable, picking up more, with less hair clog problem, etc.

Dyson DC59 VS DC35

	Dyson DC59 Animal	Dyson DC35 Multi Floor
Image
Motor	Dyson digital motor V6, provides greater suction.	Dyson digital motor V2
Dual Power Mode	Yes, the DC59 has 50% more power in boost mode than DC35.	Yes
Suction Power	28 airwatts at standard power or 100 airwatts in boost mode	28 airwatts at end of wand in standard mode or 65 airwatts at end of wand in max mode
Multiple Floor Cleaning	Yes	Yes
Cleaner Head	Wider cleaner head and is more powerful than DC35, 9.8 inch width	8.25 inch wide
Bin Capacity	0.12 gallons	smaller bin, 0.09 gallons
Bag Type	Bagless	Bagless
Battery Life	Battery life of DC59 is 26 minutes on regular speed or 6 minutes on boost.	DC35 has a 15 minute run time or 6 minutes on MAX power.
Battery Type	Re-engineered Nickel Manganese Cobalt/Li-ion transmits 1.5 times more power than DC44.	Lithium-ion battery
Charge Time	3.5 hours	3.5 hours
Docking Station	Yes	Yes
Tools Included	Mini motorized tool, Combination accessory tool, Crevice tool	Includes a crevice tool and a combination accessory tool
Cyclone Technology Type	2 Tier Radial™ cyclones, increase airflow and capture fine dust.	Root Cyclone™ technology
Warranty	2 year parts & labor warranty	2 year parts & labor warranty
Dimensions	47.8" x 8.2" x 9.8"	44.1" x 9.1" x 11.8"
Weight	4.6 pounds	4.85 lbs

Dyson DC59 VS DC44

	Dyson DC44 Animal	Dyson DC59/DC59 Animal
Images
Motor	Dyson digital motor V2	Dyson digital motor V6, enable the machine to generate more powerful suction.
Cyclone Technology Type	Root Cyclone™ technology	2 Tier Radial™ cyclone technology, increase airflow and capture fine dust.
Multiple Floor Types Cleaning	Yes	Yes
Dual Power Mode	Yes	Yes
Suction Power	28 airwatts in standard mode or 65 airwatts in max power mode	28 airwatts at standard power or 100 airwatts in boost mode
Trigger Design	Trigger of DC59 is redesigned, now it is lighter to squeeze than the DC44.
Bin Capacity	0.09 gallons	0.12 gallons, DC59 equip with larger dust bin.
Bag Type	Bagless	Bagless
Filter Design	The filter of DC59 is easier to pull out and wash.
Battery Type	Nickel Manganese Cobalt/Li-ion	Re-engineered Nickel Manganese Cobalt/Li-ion transmits 1.5 times more power than DC44.
Battery Life	DC44 has a run time of 20 minutes on standard speed or 8 minutes on Max mode.	Battery life of DC59 is 26 minutes on regular speed or 6 minutes on boost.
Charge Time	5.5 hours	3.5 hours
Location of Charging Indicator	Charging indicator of DC44 is located on the power adapter.	Charging indicator light of DC59 is on the unit body.
Cleaner Head	The new motorized cleaner head of DC59 is more effective on picking up dust and dirt across carpeting and hard floors.
Cleaner Head Width	8.25"	9.8"
Docking Station	Yes	Yes
Wall Mountable	Yes	Yes
Warranty	2 year parts & labor warranty	2 year parts & labor warranty
Weight	5.07 lbs	4.6 lbs DC59 is lighter to carry.
Dimensions	44.1" x 9.1" x 11.8"	47.8" x 8.2" x 9.8"

When we write this post (2014-02-22), we can find only Dyson DC59 Animal on dyson.com.
However, if you visit Dyson Australia official website, you can find both Dyson DC59 Animal and DC59.

image from manufacturer

Here are some similarities and difference between Dyson DC59 and DC59 Animal.

Difference:
The major difference is the color scheme, the included tool, and price.

Color:
Dyson DC59: multi-color, the part inside the dust bin is like iron grey.
DC59 ANIMAL: multi-color, the part inside the dust bin is purple.

Price:
Dyson DC59: $599.00 on dyson.com.au
DC59 ANIMAL: $649.00 on dyson.com.au

Tools:
Dyson DC59: Combination accessory tool, Crevice tool
DC59 ANIMAL: Mini motorized tool for tough tasks such as picking up pet hair and dirt from upholstery and tight spots.
Combination tool,
Crevice tool

Similarities:
Corded or Cordless: Cordless/Battery Powered
Cyclone Technology: 2 Tier Radial cyclone technology
Cleaner head: Motorized head with Carbon Fibre filaments
Motor: 350W, Dyson digital motor V6
Suction power: Dual power mode, 28 airwatts at Standard Power or 100 airwatts in Boost mode
Bin Capacity: 0.12 gallons
Weight: 4.6 pounds
Battery Type: Nickel maganese cobalt/Li-Ion, re-engineered nickel manganese cobalt battery and a run-time enhancing trigger provide more power and longer battery life.
Battery Run Time: 26 mins cleaning time (6 min on boost)
Battery Charge Time: 3.5 hours

Other features:
Balanced for floor to ceiling cleaning

Docking station for storing the vacuum for charging.
Hygienic one-touch bin empty

If you know other difference between these 2 Dyson hand/stick vacuum cleaners, please share below to help potential buyers to make a right purchase decision. Thanks.