This one is a bit easier. Shorter and wider are for whitewater and long and thin are for flat water, and all the variations in between for a little bit of both.
You know how water pressure rises the farther down you go? Well, to a small degree flat water (or lake water) paddles take advantage of this. A long thin blade gains traction on the water deeper down, where, because of the pressure, you can actually grab it more effectively. It’s like heavier water. The thinner width makes it easier to smoothly insert it into the water and withdraw it at the end of your stroke. It also just feels nice. A long Ottertail is a real pleasure in flatwater. You really have to try out both to understand.
It’s a different story in whitewater where responsiveness is key. Trying to haul the boat over with a repeated and frantic draw with a long flatwater paddle is an awkward mess. Exchange all that long surface area for a wide surface area and you can get that paddle in and out of the water effectively, but you’ve lost some of the smooth finesse of a long blade, and the bonus points from the deeper water pressure. There’s a little trade off there.
If you have to choose between the two, choose based on the whitewater. You can use a whitewater paddle in flatwater, but you can’t (or really shouldn’t) use a flatwater paddle in whitewater. You may want to just bring both, but for all the same reasons, a flatwater paddle doesn’t make for a good backup paddle if you drop your whitewater paddle in a rapid.
What about the ones in between? Sure, there are many that are just basic paddles in between. There’s a simple beaver tail with a rounded bottom, which helps the paddle smoothly enter the water, but it doesn’t get you the bonus points from the depth. Those aren’t really favourite of mine. There’s also the shape of the basic plastic canoe paddle. We have a bunch of those as loaners. They’re fine—I mean I would never use one, but it’s enough surface area to push water around in whitewater to get you through. Since paddles are so expensive, these simple plastic paddles are still what outfitters will give you when they rent you a top of the line whitewater canoe.