To simplify the given expression, we will first expand and simplify each term within the expression.

Expand (a+1)(a^2-b^2):
(a+1)(a^2-b^2)
= a(a^2-b^2) + 1(a^2-b^2)
= a^3 - ab^2 + a^2 - b^2

Expand (a+b)(a^2-1):
(a+b)(a^2-1)
= a(a^2-1) + b(a^2-1)
= a^3 - a + ba^2 - b

Now, put the simplified terms together:

(a^3 - ab^2 + a^2 - b^2) - (a^3 - a + ba^2 - b)
= a^3 - ab^2 + a^2 - b^2 - a^3 + a - ba^2 + b
= a^2 - ab^2 + a - b^2 - a + b - ba^2
= a^2 - ab^2 + a - b^2 - a + b - ba^2
= a^2 - 2ab^2 + b

Therefore, (a+1)(a^2-b^2)-(a+b)(a^2-1) simplifies to: a^2 - 2ab^2 + b.