Where did the differences originate and why?
In attempting to provide an answer to this question, I'll relate to you a story told to me as a member of a group of students wishing to learn more about Greenland paddles. Our instructor was Dave Isbell. Dave is member of Chesapeake Paddlers Association and advisor to the board of QAJAQ USA, an official club of the Kayak Clubs of Greenland. He is a long-time Greenland paddler. While describing the many different styles of Greenland paddles, Dave commented on the various shapes seen on different paddles and told a story of a conversation he once had with Greenlander John Pedersen who still hunts seals from a qajaq.
"The Inuit found long ago that when hunting seals, the drip coming from the end of a round paddle could be heard by their prey. It served as a warning that a hunter was near. To prevent this, the Inuit learned that putting a square tip on their paddle gave them much better control of the drip during the closing moments of the stalk. Accordingly, hunts became more successful and the square tip is now commonly seen on native paddles."
So, there you have it. The squared tip has a functional basis. If you answer yes to my question, "Do you plan on hunting seals with your Greenland paddle?", then you need a square tip.
On the practical side, there are other factors to be considered. Several paddlers who use paddles with squared tips have told me their paddle is noisier than a round tipped paddle. They say the paddle tends to "cavitate" in the early part of the stroke. This would argue for a round tip.
Dave Isbell notes that the squared tip stays put better when doing a shotgun roll. He prefers a squared tip with the corners radiused about the size of a nickel or quarter. So this is an argument for a squared tip.
I don’t have a sound basis for either choice and leave the decision up to the customer. When asked for a recommendation, I tactfully suggest round tips - or at least rounded tips. My personal preference, however, for purely esthetic reasons, is a square tip. Why? Because I think they look cool.