The difference between Hoover Linx BH50010, Linx Signature BH50020, and BH50020PC are as follows:

The Linx Signature BH50020 and BH50020PC are the same, the only difference is the packaging.

The difference between the original Linx BH50010 and Linx Signature BH50020/BH50020PC is:
The Linx Signature is a newer model with slight cosmetic and design change.

The Linx is in lighter color, something like glacier blue while the Signature is darker, like black. The dust cup on the Linx is also in lighter color. However, both allow you to see through to know what and how much are collected.
The Linx has a red plastic part inside the canister whereas the Signature has a clear one.
The Signature has plastic covers over the wheels on the nozzle while the Linx normal has exposed rear wheels. According to user reviews, the wheels on the Signature makes it roll better, more smoothly, and quieter on hard floors.
The features, operation, major parts of the Linx and Linx Signature, such as the motor, are the same.

Compare Hoover Linx and Linx Signature Specs Side by Side

	Hoover Linx Signature Cordless Stick Vacuum BH50020/Bh50020PC	Hoover Linx Cordless Stick Vacuum Cleaner BH50010
Design	Stick	Stick
Bagged or Bagless	Bagless	Bagless
Power Source	Battery Driven	Battery Driven
Power	18 Volts	18 Volts
Charge Time	Up to 3 hours	Up to 3 hours
Battery Type	Lithium Ion	Lithium Ion
Fully Charge Indicator	Yes	Yes
Run Time	about 15 minutes	about 15 minutes
Battery Life Indicator	Fuel Gauge	Fuel Gauge
Motorized Brushroll	Yes	Yes
Brushroll On/Off	Yes	Yes
Washable Filters	Yes	Yes
Headlights	No	No
HEPA Filtration	No	No
Weight	approximately 7.3 pounds	7.3 pounds
WindTunnel Technology	Yes	Yes
Nozzle Width	11"	11"
Height Adjustment	Self Adjust, No Manual Settings	Self Adjust, No Manual Settings
Brushed Edge Cleaning	Yes	Yes
Included Tools	No	No
Handle	Fixed Extreme Recline Handle	Fixed Extreme Recline Handle
Cleaning Wand	No	No
Clear Dirt Cup	Yes	Yes
Clear Nozzle Window	No	No
Bottom Release Dirt Cup	Yes	Yes
Hose	No	No
Color	Black	Black and Glacier Blue
Self-propelled	No	No
Low Profile Base	Yes	Yes
Full Bin Indicator	No	No
Dirt Finder	No	No
Warranty	2 year limited warranty	2 year limited warranty
Manual	download manual	User Guide

Some questions and answers:

• Can they be transformed into a hand vacuum?
  No.
• Do they have suction control?
  No suction adjuster.
• Can you turn off the brush roll?
  Yes. You can switch between Suction Only mode or Suction with Brush mode.
• Is the roller removable?
  You will need to remove screws to remove the bottom plate to access the brushroll.
• Do they come with tools like hose or wand?
  No tools included. They do not allow you to use hose, wand or other attachments.
• Do they come with wall mount?
  No.
• Do they stand upright by their own?
  Yes, they can stand alone.
• Can you remove the battery?
  Yes. They have removable battery.
• Is the battery interchangeable between them?
  Yes, they use the same battery.
• Do you have to remove the battery to charge or leave the battery installed to charge?
  You have to remove the battery to charge. There is a table top charging base.
• Do they come with charger?
  Yes.
• Can you leave battery in charger after fully charged?
  It is recommended to remove the battery.
• Can you use them with cord?
  No.
• Do they have HEPA filter?
  No.
• Is the filter washable?
  Yes.
• Do they use bags?
  No, they are bagless.

Similarities:

• Cordless, powered by 18V Lithium-Ion Battery.
• Come with table top charging base.
• Fingertip controls allow you to turn on/off the vacuum, turn on/off brushroll without bending over.
• Fuel gauge near the top shows you how much battery power remaining and when it needs to be charged.
• Feature WindTunnel Technology to help lift and remove debris and dirt.
• Lightweight, weighs under 10 pounds.
• Low Profile Design allows it to get under and around furniture more easily than bulky upright vacuum.