Regarding how to connect and measure differential signals, page 10 of the user guide here: https://www.picotech.com/download/manua ... 5-en-3.pdf
gives details on how to do this.
Regarding the considerations of using a differential scope, and whether you need to use a laptop that is connected to the mains or not, we need to be clear on what the PicoScope 3425 can do and what it can't do.
It CAN isolate the input section of the channels from the rest of the oscilloscope, and therefore from the ground reference that the PicoScope uses, so that you can make measurements that are with reference to a voltage that is not ground. This can include measurements of up to 400V, when combining the dfferential voltage measured with the common mode range being used, as long as the electrical energy that can be discharged between the circuit being measured and and the measurement equipment, would not be enough to damage the equipment. You can read more about what CAN do here: https://www.picotech.com/oscilloscope/3 ... 5-overview
What it CANNOT do is protect the internals of the PicoScope and the equipment connect via USB to the PicoScope from damaging electrical currents and high energy impulse. IF YOU ARE INTENDING TO USE THIS PICOSCOPE WITH HIGH VOLTAGE EQUIPMENT, THEN THAT EQUIPMENT MUST HAVE LIMITED CURRENT, AND LIMITED ENERGY IMPULSE CAPABILITY. So, for instance, you could measure the circuitry of Copier machines, that have high voltages, but low currents, and are not directly connected to electrical mains supplies, but are connected through isolating transformers.