Merriam-Webster's Eleventh Collegiate Dictionary (2003) provides these definitions for antedate:
antedate vt (1572) 1 a : to date as of a time prior to that of execution b : to assign to a date prior to that of actual occurrence 2 archaic ANTICIPATE 3 : to precede in time
and this one for predate:
predate vt (ca. 1864) ANTEDATE
Roughly the same definitions of antedate and the explicit equating of predate with, simply, "antedate" go back to the Fifth Collegiate (1936). This suggests that, with regard to the senses covered by the longtime definitions of antedate, there is no meaningful difference in usage.
Meanwhile, in the 70 years between 1936 and 2005, the relative popularity of the two terms changed considerably, as this Ngram chart for antedated (blue line) versus predated (red line) illustrates:
Although the decline in antedated seems to mirror the rise in predated, a confounding variable is at work: use of predated as a past tense of predate in the sense of "acted as a predator upon"—a meaning that Merriam-Webster as yet does not officially acknowledge, but that is not at all rare in recent Google Books matches such as this one, from Folia Zoologica, volume 53 (2004) [snippet]:
In 2002, 34% of the nests were predated, while this proportion increased to 76% in 2003 (X'with the Yates correction = 55.9, df = 1, P< 0.001). From the 34 destroyed nests in 2002, only 8.8% (3 nests) was predated by an avian predator, while a medium-sized mammal predated in 64.7% of the cases (22 nests). The fox was proved to have predated nests three times more frequently than the marten (23.5% versus 8.8%).
Still, predator predating notwithstanding, the frequency of antedated has fallen considerably since 1940, and the frequency of predated has risen quite a bit, so it appears that the popular preference has shifted in the past 70 years from antedate to predate.
Two years before Merriam-Webster's first sighting of predate, Westland Marston, The Family Credit, and Other Tales (1862) included this instance of antedated:
He [Percy Witham] drew forth the cheque book, and with the help of his pen-knife took out a leaf so skilfully that it was hardly possible to detect the excision. To avoid suspicion he filled up the page for an amount something exceeding his debt, antedated the cheque by a day, and made it payable to a feigned name, that of George Lucas.
But today, an author describing the same scene might be at least as likely to use predated.