This is a very specific problem, which many users may never face. But this is an example which Illustrator handles perfectly, but Designer (and Publisher too) fails to handle.
Sometimes I edit music sheets and so I need to import PDF with scores. Each score (note) is a glyph of a specially designed font. The typeface has one problem - it does not have whitespace characters.
In result the imported PDF is full of nodef glyphs. Apparently it doesn't matter whether during import I uncheck "Favor editable text over fidelity" and "Group lines of text into text frames".
Illustrator, on the other hand, produces clean result:
PDF file and the music font are attached. Font of the lyrics is copyrighted, but has no matter in this case.
ad_te_levavi.pdf GRECILIAE.TTF