You cannot say "Buy me a beverage." . A beverage can signify coffee, juice, etc. A drink is used for the "verb" form of course; and for the alcoholic beverages.

"Buy me a drink please".

"No you buy me a drink, B**tch"

In the UK 'drink' can imply an alcoholic drink like that too. It depends on the context.

Such an example would be: 'Let's go for a drink.' - implies alcoholic. 

There is one difference in America at least, drink often implies alocholic drink.  For example if someone says, "I need a drink." he is probably talking about an alcoholic drink.

I might not be right, but I consider "beverage" to be American English and "drink" British English. Just like I would say "food shopping" and not "grocery shopping".

Maybe "drink" is used often in speech, whereas "beverage" is often written on menus. I don't know if "beverage" is used everywhere, but it is in New Jersey.