To me, the major thing you need to know here is that Waltz is writing in 1981 when the world was a bipolar place and Al Qaeda did not exist. Allison, by contrast, is writing in the present era. The nature of threats looks a lot different now and that is reflected in the two men's views.
Look at the 6 arguments Waltz makes at the end of his essay. They are generally focussed on the US and USSR and they assume that states are the ones controlling nuclear weapons. He also assumes a bipolar world and that is a big part of his reason for thinking nuclear weapons are not a threat.
By contrast, Allison worries largely about Al Qaeda and, to a lesser degree, Iran and North Korea. Allison sees the potential of Al Qaeda getting a weapon as one of the biggest threats to the West.
Other things that worry Allison come out of the collapse of the bipolar world. He worries about North Korea getting enough weapons that Japan and South Korea decide that they need nukes. He worries about Iran getting weapons and causing Saudi Arabia to want them. In short, he worries about a multipolar world with no superpowers that can control the smaller nuclear powers.
So the two articles are written in different worlds and Allison, at least, would say that Waltz's view is obsolete.