For German names, this is a problem. I have an ü in my name and this is transcribed as a "ue" in my passport. Transcribing it as u would produce a different name (which AFAIK actually exists).
In Hungarian the diacritics are also important, for example Szilasi and Szilási are different and are pronounced differently. Still, it won't be an issue when flying or other stuff.
German is more complicated though with all the substitution rules.
Not to mention Germans who actually have an ue in their name, still pronounced as ü, but written as ue only, never as ü. Or someone may be called Gross, but it would be incorrect to write it as Groß, while someone else's name may be Groß with the acceptable alternative spelling Gross when ß is unavailable.
