It's not the material of the speaker (driver) that makes them sound good. That's about the only material they could make in the 50's. If they'd had poly's back then, they'd have used them - less mass to move means sound reproduced more accurately (somewhat at the expense of low-end reproduction). In almost every case, the output path makes a bigger difference to sound quality than the speaker. This refers to the pre-amp, processing (if any), and amplifier components.
If you take two 'decent' speakers, and connect them to any of three or four different amps (see below), they will sound noticably, significantly different.
In lower-quality amplifiers, reproduction of low frequencies saps most of the available power - this means the mids and highs (although easier for an amp to reproduce) get less power and end up suffering. Tube amps rated at 20 watts can rattle the art on your walls, because they can deliver a higher voltage more consistently - this used to be referred to as slew rate, now it's more commonly referred to as "damping factor".
Today's amps compensate for the lack of output voltage (which require heavy, bulky components that need to warm up in order to sound good) by jacking up the wattage. This is all fine and dandy, right until you try to listen to something critically. These days, most people compensate for this with powered subwoofers - totally defeating the purpose of stereo imaging, in my opinion. For home-theater, this is of course a moot point.
For example, I have a Yamaha RX-V995 - it's a 100 Watt RMS 5 channel amp/home theater receiver, that's showing it's age but still sounds fantastic. It's damping factor rating is 160, from 20Hz to 20kHz at 8 ohms. The comparably priced, new, Pioneer Elite VSX-81TXV doesn't even disclose this measurement. The $1,095 Onkyo TX-SR804 is rated at 105 Watts (not RMS), and has a damping factor of merely 60.
Just goes to show that it's not about how much you spend, or how fancy the name or the case looks. Not every Yamaha is created equal - you won't find this unit or it's equivalents at Best Buy in the Yamaha section.
Why is damping factor significant? Accuracy in sound reproduction. Here's what Wikipedia has to say on the matter:
In loudspeaker systems, damping factor describes the ability of the amplifier to control undesirable movement of the speaker cone near the resonant frequency of the speaker system. A speaker diaphragm has mass, and the surround has stiffness. Together these form a resonant system and the cone may resonate in response to short audio pulses.
A high damping factor indicates that an amplifier will have greater control over the movement of the speaker cone, particularly in the bass region where the resonant frequency of the speaker system will lie. This damping gives a "tight bass" sound from the sound system.
Regarding the "warm" sound of tube amps / preamps, consider the following:
"In my day, I sold a number of tube amps to musicians such as the
Grateful Dead. Usually MacIntosh 2300's. Reasons? Well, for one, tubes
emphasize even order harmonics while transistors emphasize odd order
harmonics. The result is a smoother, more hi-fidelity sound. The ear
is much more sensitive to odd order harmonic distortion than even.
Some have suggested this was a major reason the sound quality of early
CD's suffered as well.
Next, in the early days, tube amps would drive lower impedance loads
as they had autoformers at the output. They would develop a little
more power and drive more speakers consequently.
And last, in general, they had much larger capacitors which act as
batteries, particularly for low frequency. If the amp was driven to
close to full power, while the power supply was trying to catch up on
those low notes, the caps would supply the necessary power."
I love audio dearly - if I could make enough money doing it to support my family, I'd quit tomorrow.