Following is a response from DHS staff member, Tom Pullum:
The state-specific wealth index gives quintiles within the states, while the usual wealth index gives quintiles at the national level. The components and weights for the different components are the same, but the difference between the different versions is just in where the four cut points (to give five quintiles) are placed on the continuous scale. The urban and rural versions give quintiles within the urban and rural subpopulations, respectively. Please refer to the DHS Comparative Report 6 and the NFHS-5 final report for more information on the construction of the wealth index. It is recommended to use the national version for a national-level analysis or if you want to compare states, and the state version for a within-state analysis. There is information about the wealth index on the DHS website: https://www.dhsprogram.com/topics/wealth-index/index.cfm