How many times faster were the printing presses, which gave Europe an advantage over other continents in term of access to knowledge, compared to copying of books by hand?
According to Wikipedia, the average human handwrites at about 22 words per minute or 1300 words an hour when copying something. The bible has about 800,000 words and the Gutenberg has about 1200 pages so that's 660 words per page. Along that logic a person can copy around 2 pages per hour. The wikipedia entry suggests about 4 pages per hour by the way. Obviously it would be longer if you want to try to make the script look really pretty for a rich patron.
Now Gutenberg's first printing press could supposedly print about 25 pages per hour. So that's 10 times as fast. I suspect setting up the type for that page took quite a while, considerably longer than handwriting the same page would take. However, you could theoretically make up for that in volume by printing out lots of copies of that page.
Later presses of course improved on this, improving the speed of setting the movable type being a priority. Wikipedia's printing press entry has pretty good typical numbers:
Basically where throughput is concerned, the two aren't even comparable. In a world that includes printing presses, a person trying to promote an idea via hand-copying is bringing a knife to a gunfight.
By the way, the idea of copying a book page by page (so called block-printing) was already used in Asia for more than a thousand years. At the time of Gutenberg, Chinese printers where said to achieve up to 200 pages an hour. So it is inacurate to say that this gave Europa an advantage. It would take another 400 years before printing speed was significantly increased but by then the speed increases were fast and dramatic.
T.E.D.'s answer is the definitive one here, but the other thing to remember is scalability: it might take a while to set the type for a printed page, but every copy after that first one takes a fraction as long to print because the type has already been set. Copying a page by hand takes just as long the 1,000th time as it did the first, and that's not even accounting for the possibility of errors that would render a page unusable.
Thus, printing presses might NOT have been much faster for projects requiring only 1-2 dozen copies, but the benefit would grow proportionally to the size of the project. Printing jobs requiring many, many copies, like the Bible or a widely-circulated pamphlet, were able to reap not only the short-term benefits of printing speed, but the repetitive bonus of cranking out copy after copy from a single design.