The answer to this question is relatively simple. The title "symphony in D minor” helps distinguish which symphony it is – sort of like a working title, since most composers didn’t give fancy names to their works. For example, Mozart only wrote two G minor symphonies – No. 25 and No. 40. So, if an announcer presents a symphony as Mozart’s G minor symphony, you know that it can only be one of two symphonies. If a composer has written more than one symphony in the same key, the announcer should tell you which number it is.
As for modulating a D minor symphony to the key of C minor, for many people the difference probably wouldn’t be noticeable unless they were played simultaneously or back to back. However, to the more experienced listener it can mean a world of difference. Every key signature has its own mood/feel, like colors on a color wheel. And though D minor and C minor are relatively close together, the difference can be compared to going from royal blue to cerulean.
