Draughty PVC window

This is not meant to be a complete answer, but might help with your understanding of it. My understanding of the physics of windows is as follows -

Nearly all radiation (visible, UV, infrared, every other type too) that travels from outside the house into the house and is absorbed (not reflected back out again) is converted into heat. I would expect that exceptions (such as the energy from UV light causing a chemical change, such as when it fades your couch, or waves that induce a current in a conductor) are relatively negligible.

This means any radiation that can be reflected from the window to prevent it entering the house reduces the heat load. Also, if the radiation is absorbed into the window glass (eg through tinting it), then some of the heat from the window glass will be lost to the outside (through both conductance/convection and some radiated heat) which is an additional benefit.

A heated window pane would like lose more heat to the outside than the inside because the wind would help keep the temperature difference between the glass and the adjacent outside air larger than the temperature difference between the glass and the inside air.

Soundproofing as it relates to windows is all about mass -- the denser the materials, the more sound reduction you'll get. PVC is, indeed, typically more dense than the far more lightweight aluminium and so will will resist noise transfer more. However, the amount of surface area made up by the frame is absolutely dwarfed by the glass itself. Any differences in the frame will be massively overshadowed by the properties of the glass, make it a moot point.

PVC/Vinyl windows do a far better job of restricting conductive heat as they are a much better insulator than aluminium. Aluminium is really one of the worst materials to use from an insulation standpoint.

However, PVC/Vinyl will eventually "melt" in a hot environment. Recycled PVC will do so far faster than virgin PVC, but even the latter will eventually sag enough that opening and closing the window will be tricky. Assume maybe 15-20 years.

Aluminium will last indefinitely. That and the price are typically the two big selling points of that material.

In short, neither has a true advantage from a soundproofing perspective (the glass does quite a bit, though) but PVC is a better choice from an insulation standpoint if you are willing to replace the window every 15 years or so. If you don't care about the insulation value and don't want to replace the window, then aluminium is the way to go.