The non-technical term is "drunk driving." They both mean "drunk driving." So do "OUI" (operating [a motor vehicle] while under the influence [of alcohol]) and "OWI" (operating while intoxicated).
The phrase "drunk driving" is unpleasantly direct. For a famous rock star like Justin Bieber, the newspapers and Bieber's fans would use the softer phrase "DUI," while people who dislike Bieber would say "drunk driving."
It's an abbreviation of a legal term.
"DUI" is to "drunk driving" as "motor vehicle" is to "car." In each case the first is a technical, legal expression while the second is the ordinary word or phrase.
In the United States, each of the 50 states has a lot of independence, and there are large areas of the law that are governed by state law, rather than federal (national) law.
The abbreviation used almost certainly depends on the particular words that happen to be used in that state's laws.
From quick Googling, I think that in Missouri they use the abbreviation DWI, while in Florida, where Justin Bieber was arrested, the term is DUI.
You also see OUI, operating [a motor vehicle] under the influence [of alcohol] and OWI, operating while intoxicated.
In fact, according to this web page, in Alaska,
if you are breaking the city of Anchorage's laws, it is OUI, but if you are breaking the state of Alaska's laws, it is DUI!