Are there cases when they directly contradict each other, or can I just apply my Russian knowledge when writing texts in English?
In order for the question to be reasonably scoped, let's stick to the symbols listed below:
, . : ; ( ) ? ! - "
Let me start by telling the differences in comma placement.
Strunk's "The Elements of Style", 1918, suggests the following differences:
In a series of three or more terms with a single conjunction, use a comma after each term except the last.
In Russian, we do not place a comma before the conjunction:
Restrictive relative clauses are not set off by commas. Only non-restrictive relative clauses should be surrounded by commas.
In Russian, all relative clauses are set off by commas:
Do not join independent clauses by a comma. Use a semicolon.
In Russian, we can have independent sentences that are joined by a comma only:
That is all I have found so far.
The very structure of the sentence, the syntax, the intonation are different in English, at least this is a reason that punctuation rules are not 100% equivalent.
I agree with Alenanno, this topic is too broad. But shortest answer is no. No, you cannot just rely on your knowledge of English punctuation while writing texts in Russian (after all, this is Russian SE, why should we discuss English at all).
Out of the head, here are main aspects, specific to Russian punctuation:
Direct speech. In English we can write:
In Russian we should insert dash (—) as well:
Usage of semicolon. While semicolon exists in English as well, in Russian its usage is more common. This is because complex sentences (sentences with multiple clauses) are typical Russian.
Usage of dash. As well as semicolons, dashes are used for logical division of complex sentences. As with semicolons, the reason is that one can encounter complex sentences far more often in Russian compared to English. But there's yet another reason. In modern Russian we don't have so called copula verbs, so sometimes dash is used instead of such linking verb.
Usage of specific form of quotation marks So-called French quotation marks (a.k.a "кавычки ёлочкой", a.k.a guillemet) are still typographical standard in Russia. And in English, though such quotes are known, they are practically not used. But in web typography this gradually is becoming deprecated.
The rules of usage of quotation marks nested in other quotation marks differ as well.
Usage of comma in complex sentences is way more liberal in English. Compare:
Ellipsis. If you are a purist, in English you should type ellipsis with spaces. In Russian omitting spaces in ellipsis is standard. There is a difference in usage as well. In Russian it's quite often used to add ... a pause (intonational and logical) between words. In English it is, to my knowledge, less typical.
To all the nice answers here I'd like to add that punctuation in English is much more "flexible" than in Russian (as well as e.g. German). Not only are the rules (or recommendations) different between the British and American versions; they may also vary between e.g. Oxford and Cambridge versions. This leads to the fact that almost any use of punctuation can be confirmed by some dictionary and therefore people tend to write "as they feel".
To illustrate this, a case from my experience. I was asked to write an article for an American magazine, and was helped by a very good and patient section editor. We spent about a month on the article. Finally it was submitted, and the magazine editor corrected something from the version of "my" editor to how I had written it in the beginning. Despite the fact that by then I was already convinced that there was an actual rule I violated.
Another story was told to me by an English scientific writing consultant. He normally advises on the scientific writing style. But once was submitting a paper himself, to an American journal. The editor suggested the authors to "find a native speaker" to proof-read and correct their language.
Dialogues are punctuated differently in Russian and English. In Russian the dash is often used:
In English only the quotes are used for dialogues:
Since the OP states he knows Russian, I guess it's enough to answer what is the punctuation in English. I'll stick to a single aspect of punctuation and I'll consider
In English, a useful criterion to use them is to be aware of their power to separate ideas while writing. With this in mind, they can be listed in decreasing order:
As far as I know, in many cases when in Russian someone uses an ellipsis (...) in English the em-dash (—) will be used instead. Here's what Wikipedia writes on usage of the Em-dash:
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