How different are Chinese dialects, on average, relative to the differences between European languages?
Are the dialects spoken in Western and Eastern China as different as, say, Russian and Polish, or Italian and Spanish?
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Your question is an interesting one, in general how to compare the comparisons of languages and more specifically about the Chinese family.
The usual qualitative measure of difference is mutual intelligibility; but quantitative measures of this are hard to come by except by a large collection of anecdotes or great linguistic experience. That is to say there is not enough electronic material in all these varieties to do a computer comparison.
Also, labels like Mandarin, Cantonese, etc, are complicated by politics, history, culture, and translation, and may force thresholds that are in reality vague or not even there.
Most of the details of your question are answered by the wikipedia page on the varieties of Chinese languages
For most of the following there are many details left out, so consider this a broad idea with lots of small exceptions.
There are seven primary dialects of Chinese: Mandarin, Wu (Shanghainese), Gan, Xiang, Min, Hakka, and Yue (Cantonese). These dialects are all mutually unintelligible.
Within each of these dialects, there is a continuum of intelligibility, but from the extremes may not be intelligible (the wiki page gives many examples of unintelligibility within a dialect).
To say the differences are as much as the differences within say Romance is difficult. But it is often repeated that the Chinese varieties are as far apart as the Romance languages. What I think that means is that if you know one of the Chinese varieties, learning any of the other varieties is just as easy as say a French person learning Italian.
Some notes about your question:
To summarize, Chinese languages are very loosely related like Romance, for example Mandarin and Cantonese are like Italian and French, and even within Mandarin/Italian there are large differences between varieties (like Hebei vs Sichuan and Tuscan (standard) vs Sicilian). Don't take the analogy too far!