According to this MTV piece, Rare actually set out to make a straight Banjo remake, even including the same worlds as the original, but binned the idea because it "wasn't the way" the team wanted to go.
“We started off thinking, ‘Okay, let’s do almost a clever remake of the original, where we have the original worlds,’" said studio boss Gregg Mayles.
"We actually started on Mumbo’s Mountain, but all the tasks that you played in Mumbo’s Mountain weren’t quite as you expected it and different things would happen. We thought that’s not the way we wanted to go. Then we tried something else."
Nuts & Bolts was eventually born of an idea to give the player increased freedom to accomplish tasks.
“In traditional games of this genre, there’s usually a loaded task that’s created by the designer and then it’s up to the players to work out what the solution is to that task, and there’s usually only one solution. Then we thought ‘What we can we do with that? Kind of give the player more choice, kind of turn that around.’
"We actually got these quite simple tasks, but the solutions to them — there isn’t one correct way of doing it. And it came from that. I felt like it’s a different way of looking at it. Rather than [giving] Banjo and Kazooie these abilities that you learn and that are bolted on this character, can we give players abilities in a different way?”
More through the link.