List of lists elementwise greater/smaller than

Boole@Positive[a-b]

{{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}

1-UnitStep[b-a]

{{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}

kglr took the fast, vectorized operations. Here are some less optimal ones that you can have a look at for fun or for learning:

Boole@MapThread[Greater, {a, b}, 2]

{{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}

Boole@Apply[Greater, Transpose[{a, b}, {3, 2, 1}], {2}]

{{0, 1, 0}, {1, 0, 1}, {0, 1, 0}}