Well, during the intro, when everyone's hanging out in front of the castle before the M-Star crashes down, the SMB song is a bit more robust as if it's orchestrated. However, when the M-Star crashes down and everything turns into 2D objects, the song then repeats, but with a more Toy Time Galaxy-esque twist on it, more childish, more playful, in a sense. Also, slightly different notes, and different instruments. A bit more stacatto-y and jazzy, too.
For reference and recording purposes:
Mario Party 3 > Sound Room > Toad's Music Box > 10th page labeled "Story Cinema" > Prologue 2 (orchestra-sounding one) & Prologue 3 (Toy Time-sounding one)