As an adjective, aborted means terminated before completion. Abortive is an adjective meaning (1) failing to cause an intended objective, or (2) causing the termination of a pregnancy. So while abortive and the adjectival aborted share some common ground, there is a subtle distinction: Aborted implies intentional termination, while abortive, in its first sense, implies failure despite earnest effort. In the second sense of abortive, the termination may be either intentional or unintentional.
Examples
For example, the things described as aborted in these sentences are terminated on purpose:
As we know, President-elect Abraham Lincoln aborted his pre-announced public schedule in Baltimore on February 23, 1861. [Washington Post]
Newt Gingrich’s aborted presidential campaign announcement was prompted by concerns from his legal team . . . [Politico]
And in these examples, the things described as abortive fail unintentionally:
Its members still fearfully remember the bloody punishment they got after backing an abortive coup in 1993. [The Economist]
After an abortive couple of years in Hollywood, he eventually found success on stage in 1935 … [Guardian]
But even though these distinctions are useful, many writers use aborted and abortive interchangeably.