Snake-Eyes, greater agility and strength, he's a lot younger so he's got better stamina, and Snake-Eyes fought an extremely skilled opponent. From that vid, Zatoichi only fought canon fodder guards, and one guy who was a lot better in his mind than he was in real life.
True, and as for SE showing more use with the tonfa-swords, you've got to realize that he was using a katana, which is a heavy blade meant to be wielded with both hands. Zatoichi was using something like a wakizashi or a ninjato, which can easily be drawn with one hand, it's also a lot faster to swing around than a katana.