jmc

Call the primes $p$ and $q$. $1488=p^2-q^2=(p-q)\cdot (p+q)$. If $p$ or $q$ was $2$, $p^2-q^2$ would be odd, so $p$ and $q$ are both odd numbers that are less than $50$. That means $p-q$ and $p+q$ are both even numbers that are less than $100$. $1488=2^4\cdot 3\cdot 31$, so $p+q$ or $p-q$ must be divisible by $31$. The only even multiple of $31$ that is less than $100$ is $62$ so one must equal $62$, and the other must be $1488/62=24$. So, $p=\frac{62+24}{2}=43$ and $q=\frac{62-24}{2}=19$. So, the primes are $\boxed{19\text{ and }43}$.