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.
red, blue, and white
In Russian, we do not place a comma before the conjunction:
красный, синий и белый
Restrictive relative clauses are not set off by commas. Only non-restrictive relative clauses should be surrounded by commas.
The candidate who best meets these requirements will obtain the place.
The audience, which had at first been indifferent, became more and more interested.
In Russian, all relative clauses are set off by commas:
Кандидат, который лучше всего соответствует требованиям, получит работу.
Аудитория, которая сперва была безразлична, становилась все более и более заинтересованной.
Do not join independent clauses by a comma. Use a semicolon.
Stevenson's romances are entertaining; they are full of exciting adventures.
In Russian, we can have independent sentences that are joined by a comma only:
Романы Стивенсона увлекательны, они полны волнующих приключений.
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:
‘You’re right,’ he said. ‘I'm awesome’.
In Russian we should insert dash (—) as well:
"А ты прав." — сказал он, — "Я великолепен!".
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.
Я — царь!
I am a king.
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:
Я знаю, что Ольга очень строгий модератор, поэтому я постараюсь быть максимально убедителен.
I know that Olga is a strict moderator, so I'll try to be convincing :)
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.
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
. , : ; parenthesis and
- (even supposing I knew it, their full description would be too boad).
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:
.) Completely tears apart ideas. The same separation power belong to the question mark
;) It's a
, with more strength. E.g. they were divided in three groups: my sister, her boyfriend and my mother; my grandparents; my cousin and his girlfriend.
()) The content inside them can be ignored by the reader without danger.
-) They emphasize what follows, or what is nested inside dashes. Used as well to define.
:) They can be used to build lists or to define.
,) The less separation unit so to say.
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.
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:
It is also used to indicate that a sentence is unfinished because the speaker has been interrupted. For example, the em dash is used in the following way in Joseph Heller's Catch-22:
He was Cain, Ulysses, the Flying Dutchman; he was Lot in Sodom, Deirdre of the Sorrows, Sweeney in the nightingales among trees. He was the miracle ingredient Z-147. He was—
"Crazy!" Clevinger interrupted, shrieking. "That's what you are! Crazy!"
"—immense. I'm a real, slam-bang, honest-to-goodness, three-fisted humdinger. I'm a bona fide supraman."
Similarly, it can be used instead of an ellipsis to indicate aposiopesis, the rhetorical device by which a sentence is stopped short not because of interruption but because the speaker is too emotional to continue, such as Darth Vader's line "I sense something; a presence I've not felt since—" in Star Wars Episode IV: A New Hope.