They're definitely different. Good luck getting an Anvil, try the guys at Facelevel or Riverboarders of the World on FB. They've done direct Fluid orders from South Africa. The flat Kern or Earthveins boards are more diverse for surfing. I've used them all and you can do most everything with a flat board. They are quite different in weight and technique. The biggest difference is how they perform in big holes. The flat boards are harder to stay on when you get denied or don't submarine through a hole. The sleeker Anvil keeps your body closer and is more predictable when going over big drops and you can roll it easier. Check out Riverboarders of the World on FB for many opinions.
The Anvil provides more protection and seems to punch better (you can put your water bottle and other gear inside). They are heavy and you are very limited if you want to play or rescue. The Anvil makes towing a swimmer very difficult compared to the flat boards. It's made for a rental/retail market that goes for a "one fits all" approach. I've had to add stick on foam on my forearm and hip areas for custom fit. Foam boards have more float and are a bit more responsive. With all of that said I would take an Anvil over a flat board, however I keep a flat board to play on and for easier water.
I've started to make my own Hydrospeeds now. They're much lighter than the Anvil and more responsive. The Frenchies have been making them for decades. Vade Retro in France makes multiple models and designs, but are a bit industrial for most euros (good luck getting one over here). You won't go wrong with a flat board, as that's most likely what you can get, unless someone puts a used Anvil on the swap. I rode Rocky Mountain Riverboards for years and that flat board still does it all.
Anvil 10yrs later..
It's just like any other sport, you'll need to put the time in. The learning curve is easy, class I-III is pretty quick for most boaters. People start getting trashed when you step up to bigger water. You also need to be in shape, it's a bit more work than paddling. Understanding how to submarine and read water at facelevel is a little different. Buy shin/knee pads if you're in Colorado, lots of sharp rocks. I've built a few suits with armor, but you can get away with putting them under a wetsuit to start out.