I graduated from a Russian university and later taught for years at an Australian university, so I understand your question very well. In my opinion, the reason for the phenomenon you're asking about is multifold, and I will now make a few points to explain my view.
I have to start with an important caveat though. The West isn't uniform, and my answer is specifically about Australia. For a German point of view, read another answer.
1. While students at Russian universities are generally graded against well-defined standards, students at Australian universities are often graded on a curve, that is, against their peers. This means that if you cheat at a Russian university, your cheating won't affect the grades of other students, and that is why your cheating is simply none of their business. And that's often not the case at Australian universities.
In response to comments below, I want to clarify that grading on a curve is often implicit rather than explicit. When I taught at an Australian university, we first gave each student a score according to our assessment criteria, like in Russia, and then, when all scores were known, we decided how to convert these scores to grades on the scale of 100 - and the latter grades were written in transcripts. The conversion rule was chosen each time on the ad-hoc basis and had to be such that outstanding students (say, top 5-10%) got a grade above 90 (but never 100), while average students got something like 75. So, it was, in effect, a curve, but we didn't really explain this to students, so I guess that some of them were not fully aware of the competitive character of the grading process. Still, I believe that many students realized, at least on some level, that they were compared to others in the grading process and that this could affect the grades.
In contrast, it is quite normal in Russia to give every student in the class the highest possible grade, отлично - or, for example, to give most students the lowest passing grade, удовлетворительно. The point is that there are no expectations whatsoever from Russian teachers regarding the mean value and spread of the grades. Russian teachers are expected to clearly define standards before the exam and then blindly grade against them.
One more clarification: A comment below says that grading on a curve is likely to increase cheating, not the other way around, and that well may be true (provided that all other factors are the same), but OP is asking about how your cheating is viewed by others. And my point is that if grading is on a curve and you cheat, you basically push the grades of your peers down. So, I think it's understandable how they feel about cheaters - and they woudn't feel that way if grading wasn't on a curve. Also, teachers who grade on a curve have an additional motivation to ensure that no cheating takes place, to avoid fair students getting "robbed" by cheaters.
2. Australian universities care much more about their reputation than Russian universities do, because a good reputation helps attract students who are prepared to pay a high tuition fee. In Australia, students pay a lot of money for their education. Getting a BS degree in Australia can easily cost you A$100,000 (which is about US$70,000). Tuition fees are a considerable part of a university's income. And each university is free to set its own tuition fee. So, if a university gets notorious for letting its students cheat, its reputation gets destroyed, and so does its ability to attract well-paying students. After all, students want a diploma of a reputable university. And in order to remain reputable, Australian universities strive to eradicate cheating.
In contrast, education at Russian universities has been free for decades. Now many Russian universities have started introducing tuition fees, but they are not as large as in Australia, and many Russian students still pay nothing for their university education. In Russia, if you get good scores for your school-leaving exams (ЕГЭ), you will be able to choose between many universities that will grant you a tuition waiver. And if you don't get good scores, you probably don't need a university education in the first place. At any rate, Russian universities are largely state-funded, and their income doesn't depend on their reputation as strongly as for Australian universities.
So, when you cheat at a Russian university, you harm neither your fellow students, who are graded independently of your grade, nor the university itself, which gets funded by the state according to some formal criteria that your cheating has nothing to do with. So, if you don't make any obvious harm to the university or your peers, why would anyone at your university be really bothered by your cheating? No harm, no foul.
3. Academic staff at Russian universities are notoriously underpaid, with the typical salary being US$500 per month or so. They barely make ends meet, at least unless they have a second source of income such as a second job or private tutoring. As a result, they don't really value their university teaching jobs. So, they don't really care when they teach. And, consequently, they don't really care about cheating. Furthermore, if you are a Russian university teacher and let your students cheat, you'll be able to give them good grades, and that will be a good cover-up of your poor teaching - and you have to cover it up because your teaching is as good as your salary check is.
In contrast, academic staff at Australian universities are relatively well-paid. Professors in Australia make about A$200,000 per year (which is US$130,000). So, they value their jobs. They really care. And no professor wants to get notorious for letting students cheat, for no university will tolerate this - see my second point above.
4. Students in Australia tend to have a more responsible approach towards their studies as compared to their Russian counterparts, in my experience. After all, Australian students pay a lot for their education, so they naturally want to get real knowledge. And this makes them less inclined to cheat.
5. Finally, cultural differences play a role, too. Australians tend to be much less tolerant towards cheating in general (e.g., cheating on taxes) than Russians are, in my experience.
At any rate, there's no simple answer to your question. It's a very complex picture involving many factors, and I'm unsure whether I've listed all of them.
And I'm sorry if my answer seems too critical of Russian universities. I hope they are improving now, although the recent geopolitical developments don't add optimism, to be honest.