Syncronising engines in modern aircraft such as the B777 is usually a function of the FADEC (Full Authority Digital Engine Control) and the benefits are less vibration and consequently less fatigue on the structure and a quieter environment for the passengers.
However, the systems needs the engines to be reasonably matched and if one engine is changed, the parameters can be very different with the new engine being much more efficient. This is when the engines can can get out of phase with the FADEC having a hard time balancing the mismatched engines.
Older types without digital engine control need manual synchronisation. One type I flew had a 'Syncrotac' which had three little wheelts for engines 2,3 and 4. If all wheels were staionary, these three engines were sychronised with engine 1. Turboprops have a bigger problem with the propellers being larger rotaing masses. Analogue sycnronising systems were common from an ealier age, but again these were unreliable and often needed manual intervention.
Start up vibration is common. this is because the large fan baledes are slightly loose in their sockets. They rely on the enormous centripetal force when the fan is up to speed to hold them in position and give the fan it's rigidity. Before the fan is up to speed during start, one or more blade may be slightly out of position leading to the vibration.