George Clooney was originally attached to the project but when he left to do Batman & Robin, ex-Batman Michael Keaton was re-cast into the lead role. This caused no end of trouble for the Jim Henson's Creature Shop because they had designed their snowman animatronic specifically to Clooney's facial features and acting style.
When Jack and his son Charlie are riding the toboggan (to get away from the bullies), Jack breaks the toboggan into two halves and Jack and Charlie ride separate halves down the hill. When they reach the bottom, Charlie is on a real snowboard, not half of the toboggan.
When Charlie is building the snowman, he places a "The Jack Frost Band" button onto him and we see the red scarf. In the next shot, Charlie is putting the scarf onto the snowman.