Depending on where you live, the term divestment or disinvestment may have stronger political overtones to you than divestiture does. I had long been vaguely familiar with the term divestiture, but I don't remember having encountered divestment until the 1980s, when (in Berkeley, where my wife was attending graduate school) a protest movement arose over the University of California's investments in companies doing business in South Africa.
According to a Wikipedia article on the topic, the U.S. campaign against business investment in South Africa began in earnest with the promulgation in 1977 of the Sulllivan Principles and wound down in late 1989 simultaneously with the institutional demise of South Africa's apartheid system.
The popular term in Berkeley for the policy that the protesters wanted the university's regents to adopt was divestment; protesters in some other locales used the same term in formulating their demands. Elsewhere, I believe, the preferred term was disinvestment, and in some places, though less commonly, divestiture appears to have been used.
To see what sort of effect this popular movement had on the relative frequency of the terms divestment, disinvestment, and divestiture, I first set up an Ngram chart for the three words for the period from 1970 to 2008. In the chart divestment is the red line, disinvestment is the blue line, and divestiture is the green line:
Next, I added the phrase "in South Africa" to each term, using the same time period as before. Here, divestment in South Africa is the red line, disinvestment in South Africa is the blue line, and divestiture in South Africa is the green line:
As you can see, during the heyday of the movement (roughly 1981 to 1994), use of divestment and disinvestment in the context of South Africa shot up—and then declined just as rapidly—while divestiture remained the most common term overall (when context was left out of account).
So even though divestment, disinvestment, and divestiture may objectively have interchangeable definitions, I suspect that the political connotations of divestment and disinvestment from the 1980s continue to color those terms in the minds of at least some English speakers.