American Airlines Astrojet - is it really 'polished aluminium'?
Yes it is polished aluminium.
On the 737 in your photo, the weight difference with a fully painted plane is about 70 kg, and this yields operating cost savings of about USD 60,000 per year in 1998 dollars, about equal to USD 88,500 in 2016 dollars.
The paint coat is repainted every couple of years, the aluminium polish must be re-buffed 3 times per year.
From the same Boeing document:

Polished surfaces are protected from corrosion by regular buffing
after washing. Painting protects against oxidation, salts, and jet
fuel spills. However, unrepaired chips and cracks in paint collect
dirt and moisture and so may become corrosion sites. Painted surfaces
are also susceptible to filiform corrosion, or worm corrosion, which
begins between metallic surfaces and paint and erodes both. It creates
hydrogen and lifts up the paint layer as it travels across the
surface.

Also, the paint protects against small scratches from sand etc which may start fatigue cracks, however since the polished skin is very closely scrutinised during the re-polishing that is not really an issue.