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$10^{1002}-4^{501} = 2^{1002}5^{1002}-2^{1002}=2^{1002}(5^{1002}-1) = 2^{1002}(5^{501}+1)(5^{501}-1)$
Now in mod $4$,
$5^{501}+1 \equiv 2$
So it is a multipe of $2$ but not $4$.
In mod $8$ we get,
$5^{501}-1 \equiv 5\times (5^2)^{250}-1 \equiv 5-1 \equiv 4$ $[5^2 \equiv 1]$. So it is not divisible by $8$.
So the the the answer is $\boxed{2^{1005}}$