I don't think there is any difference in the way you would pronounce "man~u~fact~u~rer" or "ma-nu-fac-tu-rer" when you say them aloud.
This is mostly an issue when writing. It is a visual, publication rule. It comes into play when you need to break a word across a line and need to know where to put the hyphen. And it is only important in the most careful, formal writing. If you are writing a letter to a friend, or even a paper in a classroom, nobody is going to notice whether you write
Nowadays, in case you haven't noticed, in addition to built-in spelling dictionaries, word processors also have built-in hyphenation dictionaries. They know where to hyphenate words if they need to.
You can find rules, but none of them are reliable. When in doubt, do what native English speakers do when we are not sure how to break a word into syllables: look up the word in a dictionary. Or use a word processor and trust its automatic hyphenation feature.
By the way, according to the first dictionary I checked, the accepted syllable breaks in "manufacturer" are "man·u·fac·tur·er."