Nukes just... work. You twist a couple of keys, flip a switch and 45 minutes later human civilization ends.
A weaponised virus doesn't have these constraints, and could destroy humanity, but at least it takes some time, and can probably be somewhat mitigated.
In nuclear war, a split second decision could annihilate every living human, erase all evidence of our existence, and maybe cause a big enough climate shift to permanently end life on this planet.
Not to take too much from the argument, but that line is way overblown. I think it's hard to believe a nuclear war would kill all humans (it would certainly destroy civilization, but killing everybody is way harder) but nobody can be sure of that. Anyway, the Earth has survived meteor strikes larger than humanity's entire arsenal and life is still here.