Well, in terms of them "vibrating at the same rate for the same note," that's sort of true... A-440 is A-440, but it's not that simple. When a flute plays a given note, it clearly sounds different from when a violin plays the same note, and the reason for that has to do with several interacting factors. We usually talk about the "timbre" of a sound, which is a concept that combines the fundamental frequency with various harmonics and overtones that make an instrument sound the way it does. That A-440 (so-called "concert pitch") does indeed have a fundamental frequency of 440 cycles per second. But if ALL you heard was the fundamental, the effect is, to my ears, rather "hollow." You can get synthesizers to produce this kind of thing (and it can be good for certain lead sounds). Each resonator (which is just a word for something that vibrates) produces its own set of both even and odd harmonic frequencies, and that unique combination is what makes things "sound" different.
Also, it turns out that notes have a life span. We call the life span of a note its "envelope," and there are basically four words that describe the envelope of a note: attack (what happens right when you begin to play the note), peak (the point of greatest volume), sustain (how long the note goes on), and decay (the fading of the sound). Just as each kind of resonator makes its own unique combination of harmonics and overtones leading to its unique sound, so too does each resonator have its own characteric envelope.
Each instrument also has its own range. In this particular case, it's usually the case that the smaller or shorter the instrument, the higher the useful range. That's a little misleading given electronic instruments like virtual synths that have no "size" but almost unlimited range, but for acoustic instruments it's a general rule. An acoustic bass, with its large size and thick strings produces vibrations commonly slower than does a violin, with its smaller size and thinner strings. When you look at the registers of an organ, the lowest frequencies are set up in the large pipes, and vice versa. Looking inside a piano, an instrument with an extremely wide range, you'll notice the same thing; the low notes are produced by long, thick strings whereas the high notes are produced by shorter, thin strings. All this stuff relates to the physical characteristics of the medium being used to make the note (pitch) sound. It's easier to get sufficient volume in the lower registers with sufficient volume out of those thick strings, but trying to tune them up too high cause them to break.
Hope this helps.