DUI means Driving Under the Influence and DWI is Driving While Intoxicated or Driving While Impaired. Sometimes the term OWI (Operating While Impaired) is used. All these terms relate to driving after consuming alcohol or drugs, and are very serious crimes. The drugs don't have to be illegal for a DUI to be issued - they can be illegal narcotics, over the counter medication, or prescription drugs.
|Stands for||Driving Under the Influence||"Driving While Intoxicated" or "Driving While Impaired"|
Depending upon the state where you live, the severity of the offense may vary. In some states, the drunk driving laws differentiate between a DUI and a DWI, where the DUI is a lesser charge. In these states, a DUI usually signifies a lesser degree of intoxication, which is determined by a person's blood alcohol level at the time of arrest. Sometimes, states will allow the charges of a DWI to be reduced to a DUI. In the case of a reduction from a DWI to a DUI, certain conditions typically must be met, such as the incident being a first offense, the defendant’s display of remorse for the action, and a blood alcohol level that was not drastically over the legal limit. For example, the state of New York differentiates between DWI and DUI by establishing a blood alcohol level of .08 as the legal limit for DWI. If a person has a blood alcohol level of .07, the charges may be reduced to a DUI, which carries a lesser punishment.
Some states (like Virginia and New Jersey) do not recognize any difference between a DUI and a DWI. As far as the laws of these states are concerned, any blood alcohol level over the specified limit is a crime that will be punished in the same manner.
In Minnesota, on the other hand, there is technically no such thing as a DUI because they only use the term DWI.
In some states, the terms DUI and DWI are used to indicate whether a person was driving impaired under the influence of drugs or alcohol. In this case, DUI is reserved for illegal drugs.
The distinction for the federal government is drawn based on severity. A DWI is issued when the blood alcohol content (BAC) is over the 0.08 limit, whereas a DUI is a less severe term, given when a persons BAC is under 0.08.
edit Legal Limits for Blood Alcohol Level
In the U.S. individual states have the power to regulate DUI penalties; while all states impose penalties for driving with a BAC (blood alcohol content, or blood alcohol concentration) > 0.08, some, such as Colorado, impose criminal penalties above 0.05.
Drivers under 21 are held to stricter standards under zero tolerance laws because they are not legally allowed to drink. Adopted in varying forms in all states, these laws hold the driver to much lower blood alcohol content levels for criminal and/or license suspension purposes, commonly 0.01% to 0.05%. Federal Motor Carrier Safety Administration mandates a 0.04% BAC limit for drivers driving a commercial vehicle which requires a commercial driver's license.
Sometimes proxies such as breath PPM may be used to calculate blood alcohol content. Most states have these measures and proxies coded into the law to provide prima facie cases, just like speed limits. This means a driver measured to be over the allowed blood alcohol content limit gets automatic penalties. But even below those levels, drivers can have civil liability and other criminal guilt. For example, in some state statutes (for example Arizona), any driving impairment to any degree caused by alcohol consumption can be a civil or criminal offense in addition to other offenses at higher blood alcohol content levels.
Further reading: Legal limits for BAC - all countries (Wikipedia)
edit Video explaining the differences
Laws related to drunk driving are very state-specific. Here are some videos that explain these differences, including the one below that is from a Florida lawyer but explains the general difference nonetheless.