Arianys invested $\$ 61,000$ in an account paying an interest rate of $7 \frac{1}{8} \%$ compounded continuously. Yusuf invested $\$ 61,000$ in an account paying an interest rate of $7 \frac{5}{8} \%$ compounded monthly. After 14 years, how much more money would Yusuf have in his account than Arianys, to the nearest dollar?

Step 1 ：Given that Arianys and Yusuf both invested a principal amount of $61,000 in their respective accounts.

Step 2 ：Arianys's account has an annual interest rate of $7 \frac{1}{8} \%$ compounded continuously.

Step 3 ：Yusuf's account has an annual interest rate of $7 \frac{5}{8} \%$ compounded monthly.

Step 4 ：The time period for the investment is 14 years.

Step 5 ：First, we convert the interest rates from percent to decimal. For Arianys, the rate is $0.07125$ and for Yusuf, the rate is $0.07625$.

Step 6 ：We calculate the final amount in Arianys's account using the formula for continuous compounding, $A = P e^{rt}$, where $P$ is the principal amount, $r$ is the interest rate, and $t$ is the time. Substituting the given values, we get $A_{Arianys} = 61000 \cdot e^{0.07125 \cdot 14} = 165401.17129809465$.

Step 7 ：We calculate the final amount in Yusuf's account using the formula for monthly compounding, $A = P (1 + \frac{r}{n})^{nt}$, where $n$ is the number of times interest is applied per time period. Substituting the given values, we get $A_{Yusuf} = 61000 \cdot (1 + \frac{0.07625}{12})^{12 \cdot 14} = 176796.0178574339$.

Step 8 ：We calculate the difference between the final amounts in Yusuf's and Arianys's accounts, which is $11394.846559339261$.

Step 9 ：Rounding to the nearest dollar, Yusuf would have approximately $11395$ more in his account than Arianys after 14 years.

Step 10 ：So, the final answer is \(\boxed{11395}\).