All that extra state is there to help overlap instructions on a time scale of nanoseconds. Every instruction has its own name->register mapping. If you're switching to kernel mode or maybe if you miss a branch or for whatever other reason, the CPU consolidates you back down to running one instruction. Once that happens, you only have one name->register mapping and all the registers have their correct values. The hidden state is reduced to nothing and you only have to save the "real" state of those 16.
To expand on another answer here -- there are more values in the PRF than there are names because "old" instructions in flight can refer to outdated versions of architectural registers (the names). But a new instruction can only see the latest versions of the architectural registers, so only those 16 actually matter w.r.t. future execution: when we switch back to the task, its new instructions will ony need the 16 live values.
