The second is much more in everyday use, and it means something imaginary or fabricated. 'Factitious' which is far less frequently used refers to something artificially created, though this can be something inanimate. The example the ODE gives is of 'a factitious national identity'.
In the novel 'Billy Liar' by Keith Waterhouse, the eponymous hero has more than a fertile imagination and lives in his own world of fantasy in which he is variously playing football for England, and President of the imagined nation of Ambrosia. All this is 'fictitious'.
In the nineteenth century many nations attempted to rewrite their own histories to create a stronger sense of national identity. Many icons of history are not historical at all (See Eric Hobsbawn 'The invention of tradition' (1983)). This was 'factitious'.
Factitious is derived from the Latin facere, 'do, make'
Fictitious is from the latin ficticius 'contrive, form'
(Oxford Dictionary of English)
This was a very good question and merits a +1.