Why have one over the other? This article explains as it applies to inshore fishing.
The spinnerbait is probably the most versatile lure for bass and redfish.
Every inshore angler has one or two in their arsenal, but I don't think too many put any thought towards why spinnerbaits are designed in different ways.
The most obvious are the difference between Colorado and willow blades.
Difference Between Colorado and Willow Blades
Let's break down the physical differences between the two, then jump into their performance and which ones to select for different scenarios.
Colorado Blades

Physical Characteristics
Colorado blades are wider and shorter than willow blades. This allows deeper "cupping", displacing more water.
They can be gold, silver or painted.
Performance
Now let's look at the performance of a Colorado blade.
Pros
Because of this, a Colorado blade produces more vibration and "thump", so much you can feel it in the rod during the retrieve.
This appeals to a fish's lateral line, the line going down the side of their body that allows them to sense displaced water and vibration.

Cons
Colorado blades aren't very streamlined. Spinnerbaits equipped with one tend to roll on their side during a fast retrieve.
In all likelihood, most redfish will still hit a sideways spinnerbait. The downside is a bad hook set resulting from a hook traveling sideways through the water instead of up and down.
Willow Blades

Physical Characteristics
Willow blades are longer and narrower than their stumpy cousin.
Like a Colorado blade, they also come in gold, silver or painted colors.
Another chief difference is not their elongation, but their shape. It is shaped more like a small fish, but remember pogies are not so long and slender.
Performance
Pros
It is said a willow blade will give off more flash. I'm not sure that this makes sense, seeing a Colorado blade has just as much surface area to reflect light.
Since willow blades are streamlined, they move through water easily at high speed. Spinnerbaits equipped with willow blades are less likely to roll over.
Cons
At slower speeds, a willow blade dangles and doesn't spin as well as it could. This translates to decreased flash.
Since it's already long and slender, the willow blade doesn't vibrate nearly as much as a Colorado blade. It's even worse at slow speeds.
When to use Colorado or Willow Blades
You should use a spinnerbait when fishing dirtier water. Redfish are sight-feeders, but if the water is dirty they may have a tough time seeing a craw or swimbait.
So you want flash and vibration to get bites! Just be sure to use a slower retrieve to give redfish plenty of time to find it.
By the way... My favorite spinnerbait is Josh Hall's HD Spinnerbait. It's tough enough to handle the kind of fishing we do for redfish. Click here to read the review.

Josh Hall's HD Spinnerbait equipped w/ Colorado blade
For these reasons, I really like Colorado blades.
Now, using a spinnerbait as a "search bait" calls for willow blades.
A "search bait" is one you cast and retrieve a lot, in order to locate fish. Honestly, you don't need to catch them, you just need them to strike so you know they are there.
Once you get a strike you can slow down to give fish a lure/presentation they will commit to.
With that said, I still don't use willow-bladed spinnerbaits. Instead I use something more streamlined, like a Matrix Craw rigged onto an Owner 3/0 Flashy Swimmer, which has a willow-bladed underspin on it.

Conclusion
It's easy for anglers to fall prey to "shiny object syndrome".
You should know what you're purchasing and why it helps you out on the water, and let the fish fall for the shiny stuff!