Why does the same frequency sounds differ?

[someGorilla]

Would it not be inconsistent to use the term Harmonic sometimes and Overtone on other occasions? Where would you draw the line?

Usage, tradition, history. Musicians do call them harmonics, even if formally they are not. Study of harmonics began far before precise instrumentation was available to show the difference between an 880 Hz and an 880.03 Hz overtone. "Harmonics" was used in this sense at least from the Renaissance while a glance at a dictionary gives me "overtone" as a 19th century calque from German (so I suppose from Helmholtz). So let musicians and physicists use whatever term they like!
Of course, strictly speaking vibrating bodies don't generate exact pure harmonics.

To answer the OP, you won't get a naturally-sounding waveform like an instrument's just by adding more frequencies. Check attack-decay-sustain-release (which is however a very poor model if you want very realistical results.) For many instruments the attack phase of the sound is very important to make it recognizable. A piano doesn't sound like a piano if you cut away the attack phase (when the hammer hits the string). Try to listen to a piano recording backwards. The overtones are the same but their evolution is time-reversed, and what counts more, the attack phase is lost. You would never guess it's a piano.

 

[sophiecentaur]

@someGorilla
You make some very good points in your post and the historical note would tie in with the improvement in understanding of vibrations around that time.
I would agree with you, wholeheartedly, about pretty well all you wrote, if this were a music forum discussing harmonics. We all use the term in guitar playing. But when you look at what's said (on PF) about the 'ignorance' of non-physicists in other areas of the subject and the way the 'public' misuse all our other precious terms (force, weight, momentum - you name them), it would seem that to ignore the essential difference between the meaning of a harmonic and an overtone is to miss the point about how oscillations occur in most musical instruments. In an RF context, if you were looking for the harmonic of a known fundamental you would be using a narrow band receiver and you would be tuned to a precise twice that frequency. You would completely miss an, off- harmonic overtone. You would never be looking for it, unless the resonator you were using had possible overtone modes.