Well, see title. They allowed non-Muslims to use printing presses, so what gives?
Had the Ottomans encouraged printing, would the Islamic world have been more amenable to newly developing (Christian) European sciences?
Most likely political reasons. François I did the same thing, he banned the printing of any books in his realm in 1535, punishable by death - a policy substantially less liberal than that which the Ottomans seemed to have done. Non-Muslims, not being capable of the same political access to the center, probably didn't represent the same political threat that the mass spread of new ideas could bring, and likely there were many such groups that were more vigorously involved in commerce and hence had greater need for printing.
An alternate proposal I've seen after quick reading is that it might have been because of opposition from calligraphers. Given the political economy of the Ottoman Empire - a single empire stretching out over a highly considerable stretch of territory - this would be possible to have an impact on the actual spread of books, unlike in Europe. François's ban was of little use after all, because there were hosts of independent states beyond his control that continued to flood him with books; in the Ottoman Empire, there were no such close by centers that natively printed in the same vernacular, or furthermore high language. Thus, an entrenched faction would be able to put up much more resistance than in Europe, although that is just speculation on my part. There is mention that Murad III mighty ave allowed the sale of European books written in Turkish, Farsi, and Arabic in the ottoman Empire, but while just guessing blindly I can't help but think that printing in those languages was limited in Europe.
I had seen one site suggest that it was a cultural reason in that the Ottoman literati liked more decorations and quality that the calligraphers produced, but that seems like nonsense; Europe's literati probably liked hand made books too, but made the switch to printing. An unlikely proposal I could think of is if printing was uneconomical and wages of calligraphers lower or productivity higher enabling them to compete with printing presses, but I doubt this was the case. Perhaps a demand-side element could be at play instead though; if the Ottoman potential major-market "middle class" was of insufficient size or wealth to purchase many books, then only the upper class, already with access to calligraphers, would purchase books. In such an environment a ban on printing would be easy to continue. I don't know about the internal Ottoman economic distribution and wages to know if that is possible, but it hardly seems impossible that the Ottoman Empire was poorer than Europe and hence had a smaller market for printing.