Are there known bounds on these differences?
$378661^2-5234^3=17$ is the best I found for small numbers, which suggests to me that there's maybe a lower bound that goes as the cube root of the number being cubed, since $17^3 < 5234 < 18^3$. To be specific, I'm wondering if there really is such a lower bound, or if the difference can be arbitrarily small for large values.
On a side note, it also seems to be common to find nearby pairs with very close differences, e.g.
$3006688^3-5213538037^2=39303 \\ 5213891773^2-3006824^3=39305$
This only occurs when the mean of those differences is a cube ($39304=34^3$), so I suspect there's some obvious reason for it.