The front panel switch on the case that turns the computer on, used to be an actual switch that would make or break the current to the motherboard, some systems still have this type of switch but is now either on the PSU or at the back of the case. Modern front panel switches are now a momentary action switch ( a pulse goes to the motherboard), this press is actioned by the motherboard dependent on it's current state, IE - if off the full power is supplied to the M/B and the system boots, If the system is on and running a signal is sent to the bios, and thus the OS which actions it according to user preference, Switch off, Sleep, Hibernate etc.
I have seen "dead System" faults which are caused by a faulty switch or it's cabling to the motherboard, also there is often a four pin connector from the PSU to the M/B (in addition to the 20 pin connector that is related to this soft switching which maintains a current to the M/B when plugged into power but switched OFF
it may be worth while cleaning the contacts on this and the main PSU/ M/B connectors and ensuring that the switch connections are correct and workng