These terms are popular to describe individuals who are extreme in their smarts in respect to the social or academic realm. People tend to point these observations out on people who are at either end of the spectrum.
Street Smart individuals are those characterized by their social capabilities and heightened level of common sense. They tend to be charming individuals that prevail in social situations and have an understanding of subject matter pertaining to people's day-to-day lives. Often, these individuals grew up in a shrewd environment and tend to be less educated.
Book Smart individuals are those that are academically intelligent. The excel in the areas pertaining to academic topics and can spit out information from a textbook without lifting a finger. They are knowledgeable and eloquent in scholarly subject matter. When individuals are categorized as Book Smart it usually means they lack common sense and are clueless in social situations.
I feel a lot of people are a mixture of both with more inclinations toward one or the other. To me Book Smarts prevail in theory and often these people come from privileged backgrounds which explains their lack of experience in social situations. They have a head start and a good foundation for success. However, I feel as though in the end Street Smarts can get you further in life since a lot of upward mobility require skilled networking and being well-connected. People who are Street Smart tend to deal with many unfortunate life circumstances and overcoming these obstacles build a foundation for a success as well. Each "smart" respectively have their own benefits and skill sets.