Long version:
The old system increased the damage Illaoi gained from AD (including her base AD) If she had 100 AD, she would get 0/10/20/30/40 bonus damage depending on the number of points in Q.
The new system increases total damage by a 10/15/20/25/30%, which includes both the base damage AND the damage she gains from AD.
The damage from her tentacles without any points in Q in both systems is 10xlevel + 1.2xAD
The new AD ratio is now 1.2 x 1.1, 1.2 x 1.15, etc. which comes out to 1.32 AD, 1.38 AD, 1.44 AD, 1.50 AD, and 1.56 AD
That's a higher ratio at ranks 1-3, the same ratio at 4 rank 4, and a slightly smaller ratio at rank 5 (0.04 AD less)
The base changes from 10-180 to (10-180)x1.1, (10-180)x1.15, etc. which results in
11-198, 11.5-207, 12-216, 12.5-225, 13-234.
At level 9 when Illaoi maxes her Q, she has 117 base + 1.56 AD vs. the old system where she had 90 base + 1.6 AD. If she is below 675 AD, then she would be dealing more damage under the new system. At any other stage of the game (before 9 at lower ranks of Q or above 9 when her base damage increases) the new system is favored even harder.
The damage boost especially benefits if you build tankier rather than focusing on damage, but even the most damage focused builds for Illaoi would be gaining damage.