Teaser 2990: Squares and cubes

From The Sunday Times, 12th January 2020 [link]

Jenny is pleased that she has found two whole numbers with a remarkable property. One of them is a single digit greater than zero while the other one has two digits. The remarkable thing is that the difference of their squares is a perfect cube and the difference of their cubes is a perfect square.

Her sister Sarah is not impressed, however. She has found two three-digit numbers for which the difference of their squares is also a perfect cube and the difference of their cubes is also a perfect square.

In ascending order, what are the four numbers?

[teaser2990]