Muskoka chair vs Adirondack chair
The same chair. One name from a lake district in Ontario, the other from a mountain range in New York.

They are one chair. The tilted seat, the tall slatted back, the wide flat arms: identical. The name is the only thing that changes when you cross the border, and it changes completely — a Canadian shop that called it an Adirondack chair would confuse half its customers, and an American shop that called it a Muskoka chair would confuse all of them.
Adirondack
Thomas Lee designed it in 1903 at his family's summer place in Westport, New York, on the shore of Lake Champlain in the Adirondack Mountains. He wanted a chair that would sit level on a hillside and let a person lean back without cushions. Eleven boards from a single plank. A local carpenter, Harry Bunnell, took the design, patented it in 1905 as the "Westport chair", and built them for twenty years. The slatted back came in the 1930s and the name settled on the mountains rather than the town.
Muskoka
The Muskoka lakes, two hours north of Toronto, became Ontario's cottage country in the same decades the chair was spreading: steamships, grand hotels, then the family cottages that still line Lakes Muskoka, Rosseau and Joseph. The chair arrived with the visitors and became the chair on every dock. Somewhere along the way Ontarians started calling it after the place they sat in it, and the name stuck across most of English Canada. In British Columbia you still hear both.
What to call it
In Canada, Muskoka. In the United States, Adirondack. In a store that sells in both, like this one, whichever one you would say — the page you are reading is written in yours.
Questions
Is there any difference in the design?
None that is consistent. Individual makers vary the fan of the back or the curve of the seat, but a Muskoka chair and an Adirondack chair from the same maker are the same product.
Which name is older?
Adirondack, by decades: the chair was designed there in 1903. Muskoka as a name for it is a Canadian usage that grew through the twentieth century.