First, let's expand the expression (1+3m)(1-3m):

(1+3m)(1-3m)
= 1(1) + 1(-3m) + 3m(1) + 3m(-3m)
= 1 - 3m + 3m - 9m^2
= 1 + 0 - 9m^2
= 1 - 9m^2

Next, let's expand the expression (a+b)(a-b) - (a-b)^2:

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