Most DLSRs with "silent" or "quiet" shutter modes don't change the speed at which the shutter is operated at all. The transit time each curtain takes to traverse the height of the sensor is constant regardless of the exposure time (shutter speed) selected or if a "quiet" mode is selected. Exposure time is determined by the time difference between the movement of the first and second curtains.
What does change is the speed at which the mirror is moved out of the way of the sensor before the shot and dropped back down into the light box after the shot has been taken and when the shutter curtains are reset between the end of one shot and the beginning of the next. By moving the mirror at a slower speed, less noise is generated. By delaying the shutter reset other noise is postponed.
In some cases, such as when combined with Live View, the mirror is held up and not cycled at all. Thus no noise is produced by the mirror. The movement of the mirror is what produces the major portion of noise generated by a normal exposure taken by looking through the viewfinder.
In some cases the shutter reset is also delayed until the shutter button is released (allowing the photographer to delay the noise created by the shutter curtain reset until a time more appropriate for that noise to be generated).
In other cases the first curtain is opened at the end of the shutter reset cycle and the next exposure is begun electronically, rather than mechanically. The only noise produced at the time of exposure will then be when the curtain closes. As above, the noise created by the shutter curtain reset is then delayed until the photographer releases the shutter button.
So using or not using quiet mode will not have any effect on the life of your shutter. It might have an effect on the life of the mirror mechanism but in most cases, unless there is an event that causes mechanical damage to the mirror by something obstructing its movements or the camera is subjected to a hard bump that damages the mirror, the shutter tends to wear out first anyway.
As to which way causes more wear on the mirror mechanism it is difficult to say. The slower movement should put less stress on the parts of the mirror and sub mirror attached to it by reducing the force of the impact at the end of its travel. But the reduced voltage supplied to the motor that actuates the movement may put more stress on the motor by creating more heat per actuation. This could serve to cause the motor to fail sooner. In the real world, if you endurance tested 1,000 copies using normal mode for 500 of them and "quiet" mode for the other 500 copies you would probably see more copy to copy variation within each group than the difference between the median failure rate of one group compared to the other.