Originally (e.g., in the first edition of EGA and in Mumford's Red Book), what are now called "schemes" were referred to as "preschemes." The word "scheme" was reserved for what are now called "separated schemes." Presumably (although admittedly I am guessing here), the original idea was that all schemes should be separated by analogy with manifolds, which are by definition Hausdorff; however, unlike in the case of manifolds, in algebraic geometry one cannot introduce a notion of "separated" until the category of (pre)schemes has already been defined, hence the need for a term for "not necessarily separated scheme."
I can think of at least two possible scenarios that might have contributed to the revision of the terminology:
1) Perhaps preschemes that are not separated, or at least not easy to show separated, came up sufficiently often that people decided they should be included in the "fundamental objects of study." (If so, I would be interested to know some examples of these non-separated schemes.)
2) Perhaps, when Grothendieck and Dieudonne carefully wrote EGA so as to assume the weakest reasonable hypotheses for every proposition, it was discovered that there were far more propositions about preschemes than about schemes, and people (most notably, Grothendieck himself) decided that preschemes, not separated schemes, were the more fundamental objects.
Unfortunately, both of these scenarios are largely speculative. What actually happened? In particular, what was the motivation for the shift in terminology?