I've never heard of bringing in other kinds of drinks, like juice or soft drinks -- that really would be frowned on, I think, except maybe for a very young child. But on the other hand, I've often thought about bringing a bottle of ordinary tap water to European restaurants, since it annoys me to have to pay for a totally unnecessary bottle when obviously they have a faucet. Why that particular waste of glass and plastic isn't a bigger environmental issue in otherwise green-sensitive Europe still escapes me.
As far as liquor goes, it is indeed less common today for a restaurant not to have a license, but you still see it occasionally -- and especially when a restaurant first opens, because getting a license involves a lot of red tape.
In fact, some neighborhood restaurants continue to allow BYOB even though they also sell their own liquor. One of our favorite neighborhood restaurants (good chef, cheesy location in a strip shopping center) functioned for four or five years without a liquor license, and even though it now has one, many people still bring their own (often from the supermarket or liquor store next door). That is, they bring wine; I've never seen anyone bring beer in America, but here beer is really more of a cheap drink at parties, not something you have with nicer meals. And bringing a bottle to a restaurant seems somehow considerably less tacky than bringing a sixpack. (-;
The selling point is that the BYOB option attracts customers. These days, many people are too busy and tired to cook, so they eat out two or three times during the week. So it's just convenient to be able to choose your own wine from a wider selection, and that way you also pay only a modest corkage fee (preferably only around $5 per bottle of wine), not the usual stiff restaurant markup (often 100-150% or even more). But as others have said, not all restaurants welcome BYOB, since many rely on liquor sales for a large chunk of their revenue. So if you're interested in bringing your own wine, it's a good idea just to ask when you make the reservation.
AGB's comment about dry counties reminded me of when my parents' county was still dry, 10 or 15 years ago. To buy a drink at a restaurant back then, you had to join the restaurant's 'club' by filling out a little card with your name on it (like a frequent shopper card) and paying a nominal one-time fee, often only $1. After that, you had to show the card every time you ordered. It was really a silly hoop to jump through, which is probably why they finally changed the law. Well, that and the fact that it allowed beer and wine to be sold in supermarkets, which was also a lot more convenient.
The real underlying factor for those dry laws was of course that American society used to be much more traditionally religious. Many Europeans who get their view of the US filtered through European media (not only the popular press, also 'better' newspapers and magazines) tend to assume that all America is still a hotbed of extreme religious fundamentalism, so they may be unaware of historical changes like this one. In fact, 50 or 100 years ago even many mainline denominations did teach that drinking alcohol was a sin, but today that would seem strange in most churches, even some evangelical ones with younger memberships. In some of the more extreme cases, the ban once extended to dancing and playing cards -- now, that was really what I'd call fundamentalist. But those days are for the most part long gone -- at least, I don't know of any larger churches that still care much about those issues, though many, including my own, still oppose gambling in a more measured way.
Actually, 50 or 100 years from now the cycle may have swung back the other way, since we now know that alcohol is a key factor in heart disease, stroke, and many cancers. Who knows, it may eventually go the way of smoking, with laws passed against it for health reasons, not out of conservative religious morality. But we should note that the conservative religious morality itself grew out of very practical health concerns, at a time when alcohol addiction was just beginning to be recognized as a potentially serious social problem. Some Europeans seem inclined to lump all American religious practice under the misleading label 'Puritan,' but the teetotal movement was actually a 19th-century development, peaking with Prohibition in the early 20th century.
Anyway, it's just funny to me when Europeans are surprised at American dry laws, but not at their own blue laws (requiring stores to close on Sundays) and short weekend hours, which grew out of an equally religious mindset. Or when Europeans' public-health concerns are more about the possible hidden dangers of processed or genetically modified food than about the obvious risks of alcohol and tobacco.
That's not to say that either perspective is wrong -- probably we should all think more about all kinds of health risks, and reexamine outdated laws and seek more sensible compromises. I'm just saying that our own culture can really color how we see these issues. (-: