diesel works by compression driven auto-ignition. I.e. compressing to the point where the fuel spontaneously combusts. Hence no spark plugs like a petrol engine.
This reaction works better in a hot engine, where the heat affects the combustion reaction and the pressure required. So a diesel engine that has been running for a while burns more completely, responds quicker and the exhaust gases are hot enough to burn off residue in the catalytic converters and particulate filters on the way out. If all you do is the school run and popping to the shops it may well be a better idea to not get a diesel.*
* it may also be a good idea to not use the car for the school run at all and simply walk instead. Or if they're old enough, send them on the bus.