Let's simplify the expression step by step:

13ab(14a^2 - b^2) + 14ab(b^2 - 13a^2)

First, let's expand the terms:

= 13ab(14a^2) - 13ab(b^2) + 14ab(b^2) - 14ab(13a^2)
= 182a^3b - 13ab^3 + 14ab^3 - 182a^3b

Now, let's combine like terms:

= (182a^3b - 182a^3b) - 13ab^3 + 14ab^3
= 0 - 13ab^3 + 14ab^3
= ab^3

Therefore, the simplified expression is ab^3.