If you play the same note on a piano, organ, saxophone, guitar, or cello, you will perceive each instrument as playing the same pitch, but the personality of the sound will be vastly different on each instrument.
Instruments sound different because the relative loudness of their harmonics are different. The relative loudness of the overtones depends on the materials, size, and shape of the instrument as well as the technique of the person playing it. Some overtones may not even have any loudness at all. The relative strengths of the harmonics also change with time as the sounds reverberate and decay and resonate with each other.
Each instrument has a harmonic signature based on its unique combination of overtones. Our brains translate each unique combination of harmonics into an impression that identifies a particular instrument. These impressions may be described using adjectives like warm, mellow, bright, velvety, nasal, brassy, tender, and countless others. Our perception of this (a very subjective experience that is hard to put into mere words) allows us to recognize the “voice” of the instrument (and human voices, too!). This perception is called timbre.
Even very similar instruments can sound different, because their harmonic signatures can be different. A spinet piano sounds very different from a grand piano which sounds very different from an electronic piano. Even grand pianos sound different from each other. In fact, two pianos that are the exact same model may not sound exactly alike.