There are disposable kegs that work the way you describe - with a inner pouch that is filled with beer and gas pressure is applied to the outside of that.
For draught systems that directly push the beer, there are these gasses in common use:
- air - such as with a hand-pump
- CO2 - homebrewers and short runs
- N2 - for creamflow beers or where the beer line is longer than 50'.
- A mix of N2 and CO2
Using air to push the beer doesn't have any immediate affect on taste. However, after a few hours the oxygen in the air will start to stale the beer, causing staling flavors, and a muted hop aroma.
Using CO2 to push the beer can cause the beer to have more "fizz" and in extreme cases more carbonic bite.
For long beer lines, N2 is used to overcome the carbonation problem with CO2.
A combination of N2 and CO2, typically in the ratio 75%/25% is used for Guinness and creamflow beers. When used in conjunction with a stout or creamer faucet, this has a definite affect on the flavor and appearance of the beer - bitterness is rounded out and the beer has a thicker mouthfeel. When the beer is poured, the faucet and gas pressure cascades of bubbles to form when poured in the glass, which produce a lovely thick dense head of foam on top of the glass. With this type of serving, it's key to have a relatively low amount of dissolved CO2, or the glass will be filled with foam only.