Longevity is a mechanical problem, rather than a chemical problem (as it is in batteries). A-priori, this is neither an advantage nor a disadvantage - it is just different. But it is likely that these could live longer and be perfectly recyclable (and with no toxic materials at all).

Modern batteries are a marvel of chemistry, but they have made little progress in density in the last decades (and a lot of that progress relies on rare earth materials). There are many competitors for energy storage now, among them water fuel cells, aluminum and ultra capacitors. If these guys succeed, pressurized air will join this list as well.

I promise the LightSail offices (a former chocolate factory) are much more well lit than the article makes out. However, the editors thought it would be more dramatic -- and steam punkish. :-)

The advantage, of course, is that a Beautiful Female Inventor (TM) is pushing it. And therefore you should invest immediately. Or something like that.

In all seriousness, compressed air is a simply dreadful way to store energy. The necessity of an adiabatic cycle (store the heat from compression to use during expansion), for one thing. Tank rupture, for another.