One is metal and the other is plastic.
Neither are my first choice for siding. Both vinyl and aluminum siding will expand and contract with temperature swings. Both are designed to handle this expansion and contraction through proper installation. If you nail either too tightly to the wall you will have a problem called "oil canning" as it tries to expand and can't because it is tight against the wall. It is more pronounced in vinyl than in aluminum. The trim pieces are designed to handle the expansion and contraction of the siding.
Advantages of aluminum.
Crisper edges than in vinyl, less likely to break if struck in winter but will dent easily. Aluminum and hail don't mix which is why you don't find a lot of aluminum in areas with a lot of hail.
More color choices, bold colors aren't an issue with aluminum. they are with vinyl.
Advantages of vinyl.
Unless it is pretty cold out it is able to take more abuse that aluminum. In cold weather it can be brittle and this will become more of an issue as it gets older.
No need to paint it for years provided you like the color. It can be painted but that kind of defeats the biggest advantage of vinyl.
Disadvantages of vinyl.
Limited color palate and dark or bold colors are generally not available.
Flammable, if your neighbor has a fire expect to replace the siding on your house, best case, worst case is your house burns as well. We had a major fire in a townhouse community in Raleigh a year or two ago, dozens of homes went up with that fire because the wild fire burned up the pine straw mulch (bad idea IMHO) and caught the vinyl siding on fire. IMHO vinyl should not be used on multifamily housing.
The edges of vinyl are not crisp. I can spot vinyl on a house from the curb.
My preferred option is fiber cement siding, one name brand is Hardi-Plank. It takes paint well, has a 50 year warranty and is fireproof. We had a fire in my neighborhood and the house that burned was nearly a complete loss, the neighboring houses were undamaged by the intense flames 20 feet away.
"Pain is temporary, quitting lasts for ever." Lance Armstrong