Why can't this Asus tablet access anything over the internet though wifi is connected?
The problem, it turns out, was an IP conflict, and is related to this question.

The wifi access point decided to bestow upon itself the same IPv4 address as the WAN gateway.  So, I suppose that packets were being routed to itself which had no way to continue.

(Why it was different for each device is a new mystery. Maybe newer devices use IPv6 or some other difference in the networking code?)