Do you think there's a significant difference between buying 
a Nelo C1 with or without "wings"? Do they affect technique?

There is a difference between the two, but they will only
affect technique a little bit.  In the wider Nelo, you will
steer off the side almost every stroke, whereas in the narrow
one you will only touch the side occasionally.  Basically it's
just a preference thing.  Some paddlers find it easier to
steer in the wider ones, but as the next generation is growing
up using the narrower boats, it is our prediction that the
wide ones will eventually be gone.