smc

The prime factorizations of $768$ and $384$ are $2^8\cdot 3$ and $2^7\cdot 3,$ respectively. Note that $f(n)$ is the sum of all fractions of the form $\frac{1}{d},$ where $d$ is a positive divisor of $n.$ By geometric series, it follows that \begin{alignat}{8} f(768)&=\left(\sum_{k=0}^{8}\frac{1}{2^k}\right)+\left(\sum_{k=0}^{8}\frac{1}{2^k\cdot3}\right)&&=\frac{511}{256}+\frac{511}{768}&&=\frac{2044}{768}, \\ f(384)&=\left(\sum_{k=0}^{7}\frac{1}{2^k}\right)+\left(\sum_{k=0}^{7}\frac{1}{2^k\cdot3}\right)&&=\frac{255}{128}+\frac{255}{384}&&=\frac{1020}{384}. \end{alignat} Therefore, the answer is $f(768)-f(384)=\boxed{\frac{1}{192}}.$