I can honestly say that I never thought they would catch on, but I can also see why they would.  Just like not having to worry about learning a Darkroom helped more people use Digital, I can see how someone may not want to learn lighting to use external lights.

You shoot what you see, you don’t have to worry about ratios blindly.  It’s Shoot what you see photography, which is good for some occasions but not all.

What is Continuous Light good for?

• One or two person Studio Sessions
• Product Shots
• Short duration photography as they generate a lot more heat
• Where power is not a concern
• Distance is not great.
• You don’t have to try to over power local illumination or the sun.
• Different LEDs can be used to provide different color lights without the use of gels

What Continuous will not work for

• Mobile situations — such as speedlights
• Where you have limited power.  Even power packs only have so much voltage.
• Large group of people
• Strobes are easier to Gel most of the time.
• Strobes generate less heat
• Small flashes with AA batteries can be mounted/moved just about anywhere.
• A small external battery can give you several hours of shooting capability with a speedlight.

These are just what popped in my head.  I’m sure I missed some in both camps.