At its most basic, a DUI is an acronym for Driving Under the Influence, where as DWI is Driving While Intoxicated.
To put it another way, a DUI the intoxication maybe due to drugs, both legal and illegal; versus a DWI where the intoxication is a result of alcohol only.
The real difference really depends upon a state's legal definition of each.
For example, some states actually consider a DUI a lesser offense compared to DWI.
Often the BAC or Blood Alcohol Concentration level of the accused is what will determine if the lesser charge will be issued.
Other states don't even have a DUI charge, just DWI.
Minnesota is an example of one of these.
In those states that have a zero tolerance stance, there is essentially no difference in the charges of a DUI and a DWI.
It comes as a surprise to some that you can be charged with a DUI if you were under a heavy dose of pain killers just as easily as if you were high on illegal drugs.
There have been increasing cases of people being charged with a DUI when the intoxication was due to the effects of mixing alcohol and prescription drugs, or just prescription drugs on their own.
They may have passed the breathalyzer, but failed the field sobriety test and then arrested on suspicion of DUI.
In other words a drug related DUI does not depend upon if illegal drugs, over the counter medication, or prescription drugs were the source for the intoxication.