tests uses synthetic peptides as their test analyte, which is similar to 2nd generation tests. While 3rd generation tests uses some sort of a recombinant molecular particle as the test analyte.
From what i know, the generation wouldn't matter if the person was negative to start with, and has tested at 3 months. But it would matter alot especially for people who are truly infected. Because the increased sensitivity (the ability of 3rd gen tests to be able to detect both IgM and IgG antibodies compared to only IgG which previous generations tests for) allows for better detection ability during the first few weeks of infection.
That would allow for earlier treatment and better risk measures to be taken for sexual activities.
Thus, logically, if you tested negative at 6 weeks, you would be more comforted to know that your test was a 3rd gen instead of a 2nd gen. Of course, still, 3 months is still the period necessary for it to be conclusive.