The sustained overtones are only half the battle. Getting the attack correct (the sound profile of the first ~10 milliseconds) is really important for differentiating instruments. Plucking a guitar string and hammering a piano string have very different attack characteristics. A flute has a distinctively "breathy" attack.
Many synthesizers use a sampled recording of the actual instrument for the attack, then synthesize the sustained portion of the instrument.
Many synthesizers use a sampled recording of the actual instrument for the attack, then synthesize the sustained portion of the instrument.