Why every musical instrument sounds different than any other while listening?
Actually there are classes of musical instruments whose members will have similar sounds such as the string-class of violin, viola, cello or the class of woodwind instruments with flute,picolo,pan flute etc. or sets of drums or the class of percussions including a grand piano, stage piano, console piano etc... 

So its clear that each class has a very distinctive sound as an average person can easily classify them while within a given class it is more difficult for an average person to identify every member (but a person with a musical ear will be able to distinguish almost every type of instrument from each other, thats one of the primary differences between a musically able person from a not so one)

Coming to what makes all those instruments (both the classes and members)  sound different from each other is their physical structure and vibration mechanisms. These are specific to almost every class and to each member based on their material, geometry and driving mechanisms so that the resulting acoustical waves will have specific harmonic characteristics for each instrument (indeed this is true for any object of the nature) which will lead to distinguishable timbre / tone differences by humans.

Coming to the ability of simulating the acoustical charactheristics of an instrument by another one. There are already electronic analog/digital synthesizers which can mimic many of the instruments to some degree. There are also more sophisticated acoustical modeling programs which offer the complete re-creation of the acoustical properties of any imaginable or existing object/instrument. I'm not experienced in their results so you can look further.