There are many ways to notate different chords. I would say that it is vital that chords like A#dim/Bbdim are notated correctly.
There are two reasons I can think of off-hand for this:
1) How you name a chord is very important when you are considering its place in amongst all the chords in a particular key. I find that I encounter this when I use online transposition tools to change chord names to another key. For example: i'm playing a song in C major but would like to go up a minor third so that I am playing in Eb major. The original song has the following chords:
C Am F G7
I put in "lower by 1 tone" into the transposition engine. Unfortunately, the chords are spit out as follows:
D# Cm G# A#7
There are several problems with this result. Firstly, there is no such key/key-signature/scale as D# major. Secondly, even if there were, the suite of chords which are given are very counter intuitive: Cm has flats in it - why would it be paired with chords that are full of sharps??? Lastly, just looking at the letter names makes me think that I am playing something like this (in Roman numerals): I - bVII - IV - V This is totally incorrect and misleading.
2) The second major problem is that having the 'wrong' chord (or a chord that is 'spelled' wrong) screws things up if you have learned or are learning about voice leading. In western harmony/voice-leading, certain notes from certain chords are 'meant' to resolve in a particular way to a particular note.
For example, looking back at my previous transposition example, we see that, after the transposition, we have a final chord of A#7. There are two pretty hard and fast rules about resolving a 7th (or dominant 7th) chord: the third of the chord must rise a semi-tone to the tonic note, and the 7th of the dominant 7th must fall a semi-tone to the third of the tonic chord. In the correctly transposed key of Ebmajor, your Bb7 to Eb resolution looks like this:
Bb7 chord Eb chord (chords are shown with the lowest note of the chord closest to the bottom of the page)
Ab - - (falls a semitone) - - G
F - - - - - - - - - - - - - - - - G
D - - (rises a semitone) - - - Eb
Bb - - - - - - - - - - - - - - - - Bb
In the incorrectly transposed, non-existant key of D# major, you start to run into serious problem, and it sort of looks like this:
A7# chord D# chord
G# - (falls a semitone) - - - -Fx ('x' is a double sharp - you may see this is some works of classical music, particularly around key changes)
E# - - - - - - - - - - - - - - - - Fx
Cx - - (rises a semitone) - - -D#
A# - - - - - - - - - - - - - - - - A#
As I mentioned, there is no such chord as A#7 or D# - and you can likely see why; nobody is going to play in keys with double sharps all over the place. It makes reading the music overly complicated and difficult.
I hope that what i've written here makes a little bit of sense... I know my theory somewhat, but I rarely teach it...