Answer: 757

Small Hint:

Parametrize the solutions of $a^5=b^4$ and $c^3=d^2$

Big Hint:

Factor the resulting difference $s^2-t^4$

Solution:

Comparing prime exponents, write $$\begin{aligned} a&=t^4,\quad b=t^5,\\ c&=s^2,\quad d=s^3 \end{aligned}$$ for positive integers $s,$ $t.$ Then $$(s-t^2)(s+t^2)=s^2-t^4=19.$$ Since $19$ is prime, the factors are $1$ and $19,$ giving $s=10$ and $t^2=9.$ Thus $t=3,$ and $$\begin{aligned} d-b&=10^3-3^5\\ &=1000-243=757. \end{aligned}$$