Hi, Aleksandr! In the United States, we do say "beverage," although it's not used as often in casual speech. It's usually printed on menus, but even then, waiters and waitresses will ask us what we want for "drinks." As Peachey said, it's considered more formal; thus, its usage isn't as common as "drink" (as a noun). I hope this helps =)

They have the same meaning, but different origins. "Beverages" is a bit more formal. However, we can say "bevvies" to mean casual drinks at the pub.

It is just a case of formality really. Drinks is more casual and beverages is more formal.

In the US "drinks" often (but not necessarily) refers to alcoholic beverages. Beverages can be anything, other than water. Beverages can include alcoholic drinks too. Let's say you're in a restaurant and the waiter asks if anyone would like any drinks. He's clearly talking about alcoholic beverages. If you don't want an alcoholic beverage you might say, "No thank you. Just water for me." Or, you might say, "No, but I'll have an orange juice." So, to me, that's kind of the difference in the US. Beverages normally means non-alcoholic stuff (but again not necessarily because a menu could have "beverages" on it and include any kind of drink), but a "drink" often means something with alcohol in it. But like I said, these are not hard and fast rules, just a few examples. You can have a "drink" of water of course. But, I disagree that one word is more or less formal than the other. That's got nothing to do with it. They are both commonly used formally and informally.