Solution:

Factor out the obvious power of $2$: $10^{1002}-4^{501}=2^{1002}(5^{1002}-1)$.

Since $5^{1002}-1=(5^{501}-1)(5^{501}+1)$, and $501$ is odd, $5^{501}-1$ is divisible by $4$ but not by $8$, while $5^{501}+1$ is divisible by $2$ but not by $4$.

Thus $5^{1002}-1$ contributes exactly $2^3$, so the whole expression is divisible by $2^{1005}$ but not $2^{1006}$.

Thus, the correct answer is D .