They're closely related attempts to solve the same problem. The difference is - ironically - Laplace transforms are more rigorous, and don't leave as many loose ends.

Basically Laplace is a complete solution, while Heaviside's calculus wasn't.

It took about 100 years to work this out. Laplace's original work was early 19th century, but the transform didn't become widely used in engineering until after WWII.