Great question.
The short answer: Nothing really. When concerned with low frequency reproduction, diaphragm material is of little consequence.
The long answer: Diaphragm material will make a large difference for high frequency off-axis response, as well as wide band performance. A well damped material (such as a proplyene or paper for that matter) will transition into the distributed domain at a higher frequency, as well as the distributed effects will be less pronouced with a higher damped material.
I must explain what distributed mode is though: A speaker has two modes of operation - pistonic mode, and distributed mode. In the pistonic domain, as you can imagine, the diaphragm and moving assembly will act as a piston; moving air in a very predictable and controllable manner. Distributed mode is characterized as response (HF) attributes being largely dictated by A: Inductance (Le) and B: Diaphragm resonance and modes on the material itself. In other words, imagine distributed mode as being the mode where the voice coil's position and movement is not perfectly transmitted throughout the cone's surface. The cone itself will "bend" and wave in a chaotic and unpredictable fashion, exciting area based resonances and phase cancellations. Now remember, transducers are MASS controlled above resonance, so diaphram material deflection (spring), ductility, mass, and compliance (shear or mechanical resistance) ALL play a role in the cone's behavior. Because of the wide range of variables, and inconsistencies in materials manufacturing - predicting these behaviors is quite difficult.
As one could imagine, modern tools like FEA (Finite Element Analysis) modeling software, and experienced "know how" help engineers get closer and closer to quantifying mechanical behavior.
Anyway. Back to the question.
What's the difference? And I'm assuming here your question is in the context of MI speakers, specifically bass guitar speakers. ..
Aluminum will "break up" or enter into distributed mode much lower in frequency (if we're comparing to a paper cone for example) - due to the material itself (higher mass for given volume, denser. Has much sharper and agressive [parabolic] force/displacement curve) ductility factor. Obviously this is for a given diaphragm and dimentioned cone, and same suspension/motor.
What are the benefits/disadvantages of aluminum cone'd drivers for bass guitar?
Greater off-axis response/sensitivity (albeit, irregular and erratic) and typically better low frequency extension/reproduction (because of higher mass for given area). Overall sensitivity is a wash IMHO.
Disadvantage is some might not like the "tone" or sound. For a typical 12" aluminum driver (at least the ones I've done), first diaphragm resonance is around 350-400Hz. IIRC that's somewhere high up on the D string or G (as well as this band is produced in harmonics products from even the E and A string). To me, this sounds like a "bell" type resonance (a resonance is a transfer of two energies: kinetic and potential. or, think of mass/spring), as it's a lot of stored energy being released right there.
But really, there's no real "disadvantage". It's just different.
Personal preference.
Which brings me to another point (which is a general statement from me, not answering a question): There is no "good" or "bad", or better or best, in terms of speakers. It all comes down to the application. What are the goals and constraints (sensitivity, efficiency, size, extension) of the application? Where one type of speaker won't work well, it will excel in another application.
Couple that idea with the fact that audio and acoustics itself is being percieved by us - HUMANS! We are our own filter network. From what you see, feel, think and PAY for a product will weigh on how you HEAR a particular speaker (or bass, or amp for that matter).
Add to THAT, that the environment the speaker is placed into, 'in particular applications', can affect the total system performance (response, SPL) by up to 80%. For low frequency response (let's say below 100Hz), the room plays a HUGE roll in the sound.
Anyway. I should probably stop on my little rant. I'm getting a little carried away...
Hope this helps some.
-SDP