Both awards represent the same thing: Player of the Year.
The difference lies in who gives out the award. The John R. Wooden award is handed out by the Los Angeles Athletic Club. The Naismith College Player of the Year award is given by the Atlanta Tipoff Club. There are plenty more awards with similar titles. The Associated Press College Basketball Player of the Year award (voted on by the AP themselves), The Adolph F. Rupp Trophy (given by the Commonwealth Athletic Club of Kentucky), the NABC (National Association of Basketball Coaches) award, and the Oscar Robertson Trophy (given by the United States Basketball Writers Association).
All of these trophies/awards symbolize the same thing, but are awarded by different entities. Therein lies your difference.