The difference between being publicly intoxicated and driving while intoxicated is simple:
Public intoxication or “P.I.” is when a person who has been drinking proves to be a danger to himself or a danger to others.
For driving while intoxicated or DWI purposes, “intoxication” is defined as having lost the normal use of mental or physical faculties or the person has an alcohol concentration of .08 Blood Alcohol Content (BAC) or more.
In other words, you would have to be more intoxicated to be charged with public intoxication than you would to be with driving while intoxicated.