Persistant Staging area is used as a temporary space to hold data before it is loaded to the SAP BI system. As the operational system OLTP cannot be kept out of service for long, we use the PSA.
All the data in the source system is copied as it is to the PSA. Transformations are then carried out on PSA data before it gets loaded in the SAP BI system.
Operational Data Stores or DSO objects are used to store very detailed data or highly granular data.
This data is not suited for analysis or reporting unless the report requires detail data. DSOs can be used to take data from different sources and then combine them in a single DSO object. This data can then be transformed and sent to an InfoCube.
From SAP BI 7.0 onwards, it is mandatory to use PSA for loading data in the BI system.
So, these two perform different roles. PSA acts as a temporary buffer and ODS used to store granular data.
Hope that answers your question.