The vessel is designed for some maximum rated pressure. That pressure was chosen for a reason. Somebody (the process engineer) selected that pressure (supposedly) because at some time during start-up, or operation during upset conditions, incorrect operation, or shutting down the process, it could be possible that the vessel reaches its rated pressure. If it reaches or goes over that pressure, the vessel relief valve opens. The vessel's operating pressure, or its normal operating pressure (the pressure at which the vessel usually operates) could actually be much less than the vessels rated pressure.
If you put the valve's shutoff pressure equaling the vessel's maximum rated pressure, you are assured that the valve's maximum upstream pressure can never be higher than that, even if the valve is closed, hence that is the worst possible shutoff pressure, assuming the downstream pressure can be zero at that time.
The maximum differential pressure is the greatest possible pressure difference between the valve's inlet and the valve's outlet. That condition puts the highest fluid forces on the valve seat and seal, so opening or closing the valve is the most difficult at that time and the actuator must be sized for that condition. Usually that occurs when the valve is closed or almost closed, or is closed and you want to open it. Note that this is often the same condition for which you defined the shutoff pressure, so the maximum differential pressure for actuator operation is often equal to the maximum shutoff pressure, but it might be different.
To specify the max shutoff and maximum differential pressure you must know the worst possible combinaions of upstream and downstream startup, shutdown, normal and upset process conditions, and even maintenance conditions, so I cannot tell you what these values should be. But if you tell me that the max diff pressure for this valve is 3 bars, I could guess that the vessel's rated pressure might be 12 barg and the lowest possible pressure downstream of the valve (when that valve has to operate) is 9 barg, but that's only a guess, since I don't know your process requirements and maintenance conditions. If the downstream pressure reaches zero for some reason, the actuator will be too weak to operate that valve, so I would assume the valve would not be expected to work during that condition.
"I think it would be a good idea."
- Mahatma Gandhi (1869-1948),
when asked about Western civilization
