Repetitive occurs in a number of collocations and set terms, such as 'repetitive strain injury', 'repetitive speech', 'repetitive rhythm(s)', 'repetitive thoughts', 'repetitive lyrics', 'repetitive movements', 'repetitive work' and 'repetitive actions'. Many of those expressions are of relatively recent vintage; for both British English and American English, the prevalence of repetitive uncoincidentally rose in almost a straight line from about 1920 until about 1980, when it started to plateau or dip, whereas that of repetitious has been more or less bumping along at the same rate.
(Compare the stats here: BrE, AmE)
I'm hard pressed to think of many collocations that include repetitious: perhaps 'repetitious writing' or 'repetitious music'.
Repetitious has a distinct connotation of the repetitive action or phenomenon being tedious, or characterized by a lack of imagination, or being associated with some other obnoxious quality. Repetitive can also imply this, but in general it is used to describe a much broader range of situations.
This, in my view, is the most likely explanation for the greater prevalence of repetitive: there are simply many more occasions in which it is the most appropriate word.
You might wish to compare the multiple dictionary definitions aggregated by Thefreedictionary.com:
With regard to your question about the British English decline in the use of repetitive that you found in Google Books from 2002 onwards, I should point out that the drop between 2002 and 2008 (the last year for which statistics are reported) is only about 9%. Also, the sample is small, the period covered is brief, and the data set is not necessarily reflective of British English usage as a whole, because it excludes newspapers, magazines, online usage and speech (to mention some important categories). So I think not too much weight should be placed on this anomaly, which is in any case relatively small; I don't think it particularly qualifies as a 'dramatic turnaround'.
Similarly, one should be especially careful about drawing conclusions from the Google Books data for texts dating back to the 18th century. Not only are the same categories omitted as with the British English data, but few texts from that era have survived into today and have thus been available for inclusion in the data set. A small number of texts containing an unusually high (or low) number of occurrences can skew the data significantly if the prevalence rate in those texts lies at either end of the distribution curve for the actual prevalence as it was at the time.