It's more capacity than just pumped hydro of identical head x volume: even if we ignore the "compressed spring" of the air involved, charging would require at least the energy for lifting the water plus the energy for heating up their heat source. If you close a valve in the water connection, then pop open the air pressure vessel letting all energy still in the compressed gas you to waste, you'd still have the heat storage, and you could let the water rush into the lower reservoir through a turbine, harvesting the entire energy contained in the water (minus inefficiency). So it's definitely more, the heat and the gas discharge we just wasted.

The question is how much more, whether the symmetry suggested by the balanced state implies 2x energy storage or not.

Another mental model to further separate the two storage media (water lift and air compression) is this: you could charge separately, first raising the water with a pump, with the dry cavity at ambient pressure via an open connection to daylight, then seal the cavity an charge it as a compression vessel in the non-isobaric way. I still can't decide between trusting the "balance implies 2x" intuition or not, but it should be easy enough to calculate. At least the thought experiments show that it can't be 1x the energy stored by lifting water alone.

I consider starting with "let's do compressed air energy storage" as a given, the "do it by pumped hydro" does not come into play because pressure vessels are hard, it comes into play because building compressors and turbines that have good efficiency over a wide range of pressure is hard.