I think the thing that is worth stating is that the Sinitic languages do not share a writing standard. They share a writing system, but the 'shared written language' that is often referred to is actually more of a mutual use of a Mandarin-derived form of written Chinese as a standardised written lingua franca. What these languages generally lack is a standardised system of rendering themselves: Cantonese stands out as just about the one Chinese language with a basically fully-developed, but technically non-standardised system of rendering the spoken language as text. This answer by /u/Keyilan gives examples of the same sentence in three Chinese languages – Hakka, Wu, and Mandarin – which shows that if written as spoken, they are very different indeed. (For further context, in my native Cantonese the sentence might be 你知唔知，佢係邊個啊？我唔知。)

For a fuller overview of the history of Chinese standardisation, see keyilan again in this answer. But in short, the premise of the question (as others of keyilan's past answers have noted) is incorrect. The Chinese languages do not share a written standard, but rather most speakers of Chinese languages happen to have been educated in the use of a Mandarin-based written lingua franca, making it less of a unique case as such. The thing that would sort of need explaining is the absence of individual written standards, but the linked answers should provide context for that.

I'm interpreting your question as "why do European language look more distinct than Chinese languages when written down".

To begin with it's worth noting that there are pretty big differences within the writing of Chinese varieties - differences in syntax, and using words with different origins would result in a difference in the written forms between two varieties.

But the Chinese writing system obscures pronunciation - two varieties may pronounce the word for a concept differently, but if the two words have the same etymological origin, they will use the same character. (c.f. Mandarin ni and Cantonese nei - both written 你).

If we used a similar writing system for Romance, differences would be obscured. Consider the first line of the Lord's prayer in Portuguese and Italian:

PT: Pai nosso, que estais no céu, Santificado seja o Vosso nome

IT: Padre Nostro, che sei nei cieli, Sia santificato il tuo nome

If I wrote the Portuguese with the same grammar and word choice, but used Italian cognates (i.e. obscuring the pronunciation difference between the two languages), we would get this:

PT: Padre Nostro, che stai nei cieli, santificato sia il vostro nome

IT: Padre Nostro, che sei nei cieli, Sia santificato il tuo nome

So there's minor lexical differences (sei vs stai having different origins, as with vostro and tuo) and different syntax (sia santificato vs santificato sia), but once the pronunciation differences are obsucred, the sentences appear extremely similar. This is the effect the Chinese writing system has - it makes very different varieties appear quite similar written down (and this is ignoring the fact that generally in non-Mandarin speaking areas, the written form is still Mandarin - i.e. the local variety just isn't written down).

I totally, totally do not understand the question. I had a long response written, but after going back two or three times after writing it, I realised I'd gone off on multiple tangents just trying to get my head around what you wanted to know.

Are you asking why Europeans never implemented a single alphabet to underpin all of their languages?

Or are you asking why Europeans sharing the Latin alphabet diverged to include unique characters (e.g. the Spanish ñ, the Scandinavian ø, the German ß)?

Either way, I should point out the Romance languages are not dialects of Latin. They are descendants of Latin, bearing very little resemblance to their ancestor, and differing by degrees between one-another.