@all: Lovely discussion so far, with some nice points.
I might contradict most of you all in stating: all in all, there isn't so much difference between European and US zoos -or most of the other zoos all over the world.
They are all institutions keeping and displaying wild (and exotic) animals to the (mostly paying) public. The conditions and problems they have to deal with on a daily basis (may they be of financial, legal, climatic, biological, veterinary...nature) are pretty much the same worldwide (i.e. a tiger is still a tiger and has still "tigerish" needs and features, no matter whether it is kept in Detroit, Hong Kong or Wroclaw).
However, the conditions of and in the zoos can differ due to local conditions (Polar Bears kept in the tropics vs. Polar Bears kept in Alaska), and are mostly influenced by
a) the general attitude of the local society torwards animals and their husbandry (see the "cruel Asians" debate),
b) knowledge and competence of the people in charge
c) money.
A zoo in a Third World country or a tiny zoo somewhere in rural Iowa or Lituania will thus have problems to establish and maintain a gorilla exhibit equal to that in the Bronx Zoo.
Nevertheless, I would be careful to categorise zoos according to nations ("German zoos are like this, UK zoos are like that"...), as previous posts here in this thread fail to notice that there are more than enough exceptions to the rules. Just lumping the few zoos you have been to once (or seen in the Gallery) together, without knowing the larger picture, won't deliver an adequate image.
For example, neither are all European zoos "huge" (think of old inner city zoos like Frankfurt or the already mentioned Antwerp), nor do all American zoos sport immersion exhibits. And I think that the average zoo visitor in Europe is about as impatient as the American one when it comes to looking for hidden animals-just as the visitor remarks are "witty" and "thoughtful" both here and there.
If we look at the history of zoos, we can see that there have been several zoo "trends & fashions" since the late 19th century, when the first zoos in the modern sense (not counting Vienna or Paris) were founded in Europe and the USA. Some of these new trends could establish themselves, some not. All in all, they were supposed to make the zoo in question more profitable and competitive (by getting in more visitors, letting animals live longer etc.), both in Europe and the USA.
The exchange of these trends and ideas has always been of a mutual, international nature (even during the Cold War) that still continues. Advances, discoveries and gained experiences on the fields of (zoo) biology, veterinary medicine, zoo architecture, technology, new materials, a changed visitor attitude, changes in the legislative and finances...further this on and on.
One could say that Carl Hagenbeck imported the idea of panorama exhibits to the US zoos. Now a few major American zoos have become the outriders in some aspects of modern zoo husbandry and have imported their idea of "immersion experiences" since the 1980s/90s both nationally and internationally. You could almost call it a circular continuation...
Given the individual skill and knowledge of the designers in charge, their more or less adequate use of fake rocks, trees, animals, rusting cars...etc., often combined with a very commercial theme-park atmosphere, has been picked up by other zoos in the world-and just as in the USA, with some good, and some bad results. In Europe, the Dutch appear to have been the first in doing so, and in recent years some other European zoos, may they be in Germany (Hannover, Gelsenkirchen...), Denmark, Spain, Portugal, Switzerland...etc. jumped on the bandwagon.
"Nihil novi sub sole"-Nothing new under the sun (Ecclesiastes 1:9) also rings true for the zoo world. Fake "exotic ethnographical" zoo buildings and exhibits, like "Egyptian temples" for giraffes or "Hindu shrines" for zebu cattle were once popular in the early years of modern zoos (think of the elephant or ostrich house in pre-WWII Berlin); now "African villages" or "Hindu temples" are popping up all over again in both American and European zoos. Trends come and go...
Equally noticeable on both sides of the Atlantic is a shift torwards a more uniform species collection and presentation since the 1980s, due to CITES regulation, breeding programs, and said commercialism, most often based on rather crude zoogeographical themes-like Africa = "Savannah" = Lion, Flamingo, Meerkat, Zebra, Asia = "Jungle" = Tiger, Elephant, Reeve's Muntjac, Red Panda... etc etc.
This can be observed in both American and European zoos, and I can't say I'm all too pleased about it. Some readers might remember my comparison to commercial supermarket chain stores, and given that especially in the USA, more and more uniform exhibits are designed by the same few specialised agencies, this tendency torwards conformity seems to increase. However, I'm glad to see that there are still a few zoos (like the one in Plzen) successfully offering resistance.

BTW: "Modern" does not have to mean that something is better - nor is it going to stay that for long...
@reduakari: Can an exhibit really fulfill all demands? I somehow doubt it; in one aspect or another, you will encounter a disadvantage for at least one party involved. Naturalistic exhibits can be more labour intensive and make the prophylaxis and treatment of certain diseases (or pests) more problematic, sometimes even impossible.