Your question depends a bit on what you mean by "printing," and the answer is a little bit more complicated than a simple "yes or no." It has, in part, to do with issues outside the simple invention of moveable type itself, especially with regard to the respective alphabets used by China and later, by the Holy Roman Empire, where Johannes Gutenberg, developer of the printing press in Europe, lived.

China

First, moveable type -- that is, pre-created typographical elements that can be set and used to press a message into a vehicle (usually paper) using an indelible medium (usually ink) -- was indeed invented in China. The inventor and somewhat mysterious figure Bi-Sheng developed the technology, some time between 1041 and 1048 AD, according to the historical essay collection Dream Pool Essays by contemporary scholar Shen Kuo:

Bi Sheng, a man of unofficial position, made movable type. His method was as follows: he took sticky clay and cut in it characters as thin as the edge of a coin. Each character formed, as it were, a single type. He baked them in the fire to make them hard. He had previously prepared an iron plate and he had covered his plate with a mixture of pine resin, wax, and paper ashes. When he wished to print, he took an iron frame and set it on the iron plate. In this he placed the types, set close together. When the frame was full, the whole made one solid block of type. He then placed it near the fire to warm it. When the paste was slightly melted, he took a smooth board and pressed it over the surface, so that the block of type became as even as a whetstone.

For each character there were several types, and for certain common characters there were twenty or more types each, in order to be prepared for the repetition of characters on the same page. When the characters were not in use he had them arranged with paper labels, one label for each rhyme-group, and kept them in wooden cases.

-- Needham, pp. 201-202

Hopefully, this fits with what you're thinking of as "printing."

Moveable type is considered one of the "Four Great Inventions" (四大发明) of medieval China, that Chinese students learn about from a young age, so it is no surprise that your friend was familiar with it.

Europe

However, while moveable type was invented and improved in China (even to the point that early printing presses were developed around the middle of the 13th century) the ancestor of mechanical printing presses and modern printing as we know them were developed in Europe by the goldsmith Johannes Gutenberg, which he based on existing metal presses.

Gutenberg did not exactly "invent" the printing press -- like many innovators, he took existing technologies (and there were descriptions of moveable type from China and Korea that had been brought back to Europe by Gutenberg's time, though it is not clear if he would have seen them), combined and improved them, and eventually came up with something new. However, by far his most important innovation was the development of hand-molded metal printing matrices, which had not yet been developed anywhere else in the world. These allowed for the rapid creation of letters and numbers to be placed in easy settings in the press, and rearranged quickly -- further, the fact that they were cast in metal, rather than carved from porcelain or wood as in East Asia, meant that they were durable and precise, so that many printings could be made before they had to be replaced -- an efficient cost saver compared to other materials. Gutenberg used a special alloy that he invented to cast the matrices, and that alloy (tin, lead, and antimony), remained the standard for the casting of printing matrices for centuries.

The innovation of a modern, reliable printing press and the hand-molded, metal letter matrices (along with reliable inks), hugely reduced the cost of creating books and other documents in Europe. This caused printing to explode there, as supply began to drive demand.

Alphabets

A key difference in the progression of the printing press in Europe and China was the respective writing systems of the two regions. Apart from European advancements in metal casting, China had devised all elements of the technology much earlier than the Europeans. However, the relative simplicity of the Roman alphabet meant that printing large, preset pages was easy and efficient, whereas for the Chinese alphabet, woodblock printing remained more efficient due to the thousands of characters and complex arrangements thereof. (Beckwith, pg. 181) The Chinese had much of the technology backbone for a printing revolution centuries before Europeans did, but their writing system itself was not as well suited to the mass production techniques developed by the Europeans.

Therefore, the answer to your question isn't so much why the Europeans didn't adopt the Chinese system (they eventually did, in their own way), but why the Chinese did not adopt the European system -- or more accurately, beat the Europeans to it. The answer is in the respective writing systems.

EDIT: To expand on this, let us look at the history of printing in East Asia. Xylography (woodblock printing methods) continued to be by far the most popular method of printing in China and Korea, even for hundreds of years after the advent of Bi Sheng's ceramic method and others like it. Why was this -- why were more permanent methods, like injectable metal printing molds, not developed and used more commonly? Several historians, including Jacques Gernet in his book: A History of Chinese Civilization and Christopher Beckwith (cited above, who builds on Gernet) argue that the Chinese alphabet itself is one of the critical reasons -- though not the only one.

An important thing to understand about the Chinese alphabet is that it is not fixed. Centuries ago, as today, it was possible and not wholly uncommon to create new characters as they became necessary to express a new concept or describe a new thing. This makes the alphabet a constantly fluid and fluctuating thing, unlike the Roman alphabet which (depending on which particular language variant you're using) is more or less locked.

For example, looking at various historical Chinese dictionaries of characters, we see the growth of the character count from about 10,000 in 100 AD to more than 106,000 in 2004. In the 11th century alone, three different dictionaries written within 50 years of each other list character counts of approximately 26,000, 31,000, and 53,500.

This huge amount of change and pure scale of the number of characters seems to have kept xylography the printing method of choice, due to its flexibility, expandability, and cost. There was also some endemic intertia here, as the xylographic techniques of Chinese woodcarvers appear to have been quite efficient and cost-effective vis a vis metal or ceramic print making; part of this may have been an issue with simply storing and cataloguing tens of thousands of ceramic molds for use when required -- woodblock carvers could simply make new ones as needed to fill in a rare character here or there, without derailing the entire process. Please see /u/svendskov 's awesome post below for more info on how this worked.

However, even woodblock printing was not the most efficient option for the ever-changing and wide-ranging Chinese alphabet, as historian Endymion Wilkinson alludes to in his Chinese History: A New Manual:

As a result of block-printing technology, it became easier and cheaper to produce multiple copies of books quickly. By the eleventh century, the price of books had fallen by about one tenth what they had been before and as a result they were more widely disseminated. Nevertheless, even in the fifteenth century most books in major libraries were still in manuscript, not in print. Almost to the end of the empire it remained cheaper to pay a copyist than to buy a printed book. Seven hundred and fifty years after the first imperially sponsored printed works in the Northern Song, the greatest book project of the eighteenth century, the Siku quanshu 四庫全書, was produced as a manuscript, not as a printed collection. About 4 percent of it was printed in movable type in 1773, but it was hand-carved movable wooden type. Indeed, the entire collection was only printed for the first time in the 1980s. Access to books, especially large works, such as the Histories, remained difficult right into the twentieth century.

To be sure, other issues kept printing from flourishing in the Sinosphere. A sense of elitism about calligraphy versus printing for the wealthier book buyers and connoisseurs in Chinese and Korean society kept printed books from achieving cultural status. There is also evidence that Korean nobility kept a monopoly on metal printing techniques and prevented them from proliferating among the people, preventing the press from being widespread even after the adoption of Hangeul as the official alphabet of the Kingdom.

While European printers may not have faced similar elitism or constraints on their access to print, a major advantage was their fixed, simple alphabet that lent itself to durable, reusable, and easily stored and catalogued molds.

Sources:

• Needham, Joseph - Science and Civilisation in China: Paper and Printing (Cambridge, 1985)

• Beckwith, Christopher I. Empires of the Silk Road: A History of Central Eurasia from the Bronze Age to the Present (Princeton, 2009)

• He, Zhou "Diffusion of Movable Type in China and Europe: Why Were There Two Fates?" International Communication Gazette, 1994

Since movable type printing has already been discussed in this thread, my post will focus more on the history of woodblock printing in China. Due to the invention of block printing, Chinese print culture emerged several centuries before Gutenberg's lifetime. In fact, block printing arguably had a greater impact on China than movable type.

The development of printing in China can be traced back to the Tang dynasty (618–907). The exact dating for the origin of woodblock printing is somewhat controversial, ranging from the 6th to 9th centuries depending on the source. The earliest surviving block prints are sutras from the 8th century that date from the reign of Wu Zetian, which were found outside of China proper but are believed to have originated from the Tang capital.

By the 9th century, we begin to see printed materials in Tang China covering a much broader range of topics. The oldest surviving printed calendars, dating from 877 and 882, were excavated in Dunhuang. Liu Pian, a Tang official, reports observing books about geomancy, astrology, lexicography, and divination being printed during his stay in Sichuan around 883. Thousands of copies of a work on alchemy by Tang official Hegan Ji were printed starting 847.

Printing continued to flourish under the Five Dynasties period (907–960). Printing centers were established in Luoyang, Kaifeng, Nanjing, and Hangzhou. The first printed publications of Taoist and Confucian works also date from this period. The oldest Taoist work to be printed is a commentary by the Taoist monk Du Guangting dating from 913, and the Confucian classics were first printed in 953 under the supervision of prime minister Feng Tao through the National Academy.

Chinese print culture would reach greater heights under the Song dynasty. Printed books were produced in thirty locations during the Northern Song (960–1127), but by the Southern Song (1127–1279), they were produced in nearly two hundred locations, despite the Song dynasty having lost a large chunk of their territory to the Jurchens. This increase in the production of books benefited from government sponsorship and a growing commercial book industry. Printed books were now available in every subject imaginable, including works on medicine, agriculture, mathematics, dictionaries, literary anthologies, geography, novels, dramas, philosophy, and so on.

The Song dynasty also witnessed major technological advances in printing, most notably the invention of movable type by Bi Sheng. Bi Sheng used earthernware type for printing, which while notable for being the earliest known form of movable type that predates Gutenberg by 400 years, was overshadowed by Wang Zhen's invention of a practical wooden movable type. Not only did Wang pioneer wooden movable type, he also was responsible for mechanical innovations that significantly improved the speed of typesetting. Wang describes the process of producing wooden type in his Nong Shu, a treatise on agriculture:

Now, however, there is another method that is both more exact and more convenient. A compositor's forme is made of wood, strips of bamboo are used to mark the lines and a block is engraved with characters. The block is then cut into squares with a small fine saw till each character forms a separate piece. These separate characters are finished offwith a knife on all four sides, and compared and tested till they are exactly the same height and size. Then the types are placed in the columns [of the forme] and bamboo strips which have been prepared are pressed in between them. After the types have all been set in the forme, the spaces are filled in with wooden plugs, so that the type is perfectly firm and will not move. When the type is absolutely firm, the ink is smeared on and printing begins.

Metal movable type was first popularized in Ming dynasty China during the late 15th century. By then, Gutenberg had independently devised his own printing press. So what made Gutenberg's press distinct from Chinese movable type printing? /u/TheJucheisLoose has already mentioned the differences between the Roman alphabet and Chinese characters. Another is that Chinese movable type printing involved placing paper over the type and then "brushing" the ink onto the paper, whereas the Western printing press was a more efficient screw press.

In general, woodblock printing was more popular than movable type printing in China due to the financial investment of creating types for thousands of Chinese characters. This meant that movable type printing was typically limited to government-funded publications, because the costs were too high for private print shops. Another advantage of block printing was that block carvers did not need to be literate. They could carve around the outlines of characters without having to understand their meaning, whereas a typesetter had to at least know the alphabet in order to assemble lines of type. Woodblock carving also didn't require the metalworking skills necessary for producing metal type. Furthermore, after the blocks are carved, they could be used repeatedly with a negligible amount of additional labor costs. To quote the Jesuit missionary Matteo Ricci (1552–1610):

Their method of printing has one decided advantage, namely, that once these tablets are made, they can be preserved and used for making changes in the text as often as one wishes... with this method, the printer and the author are not obliged to produce here and now an excessively large edi- tion of a book, but are able to print a book in smaller or larger lots sufficient to meet the demand at the time.

There were a few cases where movable type was a popular method for printing. A significant percentage of genealogies were produced using movable type, because the content typically consisted of a few distinct characters used repeatedly. Even then, a survey of Qing dynasty genealogies shows that 58% were block prints, 34% were manuscripts, and 18% were produced with either wooden or lead type.

For further reading, see:

Tsien, Tsuen-hsuin and Joseph Needham. Science and Civilisation in China, Volume 5: Chemistry and Chemical Technology, Part I: Paper and Printing. New York: Cambridge University Press, 1985.

Brokaw, Cynthia. "On the History of the Book in China", Printing and Book Culture in Late Imperial China. Berkeley and Los Angeles: University of California Press, 2005.

Carter, Thomas. The Invention of Printing in China and its Spread Westwards. New York: Columbia University Press, 1925. (This book is nearly a century old and outdated due to the work of Needham and others, but it was highly influential as one of the first Western works to thoroughly examine the history of Chinese printing. Amusingly, it was through Chinese translations of Carter's book that first exposed China to the importance of Gutenberg in the West.)

As a follow up question, was the printing press the result of a series of other technical obstacles being conquered? Was it's invention "in the air" similar to how Leibniz and Newton both independently discovered calculus?