Answer: 330

Solution:

We must count pairs of elements of $B$ differing by $1,$ say $m$ and $m + 1.$ Adding $1$ to a binary number turns its trailing block $011\cdots1$ (a zero followed by $k$ ones) into $100\cdots0,$ changing the number of ones by $1 - k.$ Both numbers have exactly eight ones precisely when $k = 1:$ $m$ ends in $01,$ $m + 1$ ends in $10,$ and the two numbers agree everywhere else.

The shared first eleven digits then consist of the remaining seven ones and four zeros, and since leading zeros are allowed, every arrangement gives a valid pair: $\binom{11}{4} = 330.$ Each pair produces the answer $1$ exactly once, so the count is $330.$