Question one: how does bottled beer differ from draught beer?
This varies from beer to beer, and should be handled on a beer-to-beer basis. Sometimes beer in the bottle is pasteurized, while the keg is not. Sometimes one or the other is filtered, while the other is not. The gas content can also differ, since this is adjustable in draught systems but not with bottles.
In some cases, such as when yeast content is very important for the flavor profile of a beer, like German Hefeweizen, how much yeast you get in your glass has a big impact on the drinking experience. When drinking from the bottle, you can make sure that you get all of the yeast from the bottom of the bottle into your glass by agitating the last few ounces of beer, but this isn't possible with a keg.
Kegged versions of these beers still have yeast in them, but you're not getting that perfect ratio in every glass. In my two years as a beer enthusiast living in Germany, these beers are almost always served from the bottle, and every bar or bartender has their preferred method for getting all the yeast out of a bottle.
Question two: What is done to bottled beer to prolong its shelf life?
This also differs from beer to beer. Some imports are pasteurized in the bottle, some others, namely MillerCoors products, use proprietary hop derivatives like Tetrahop to keep hop flavor while minimizing hop oils' susceptibility to producing off-flavors due to light exposure. This isn't a bad or unnatural thing, and I hope this becomes more widespread, because skunked, lightstruck beer is awful. (see this interview) This, however, is also present in kegged beer, as it has other beneficial properties like increased head retention.
Sometimes, though, nothing is done beyond packaging, like Sierra Nevada's transition from twist-off to pry-off caps a few years ago to combat leaking, infection, and oxidation. Brookston Beer Blog 2007
Question three: Why is draught so much better?
Very, very subjective, and you may want to clarify or qualify this in the future. Is good draught beer better than old, skunked bottled beer? Yes. Is good bottled beer better than flat draught beer poured through dirty lines? Also yes. Each has its pros and cons when it comes to service and storage.
I would venture to say that much of draught beer's appeal comes from it being special- you typically have to go somewhere else to get it in a way that's not usually available to the home user, like how the theater experience differs from the home movie experience. Yes, you can get bottled beer in bars, too, but it's still a more "normal" format. In terms of taste over experience, though, I don't know that you could say for sure without some sort of blind tasting with lots of variables (age, glass type, line cleanliness, CO2 pressure) controlled for. Even then, what does it for the tasters might not do it for you.
For further reading on all that goes into draught beer, check out the Draught Beer Quality Manual