Specifically what species do we see now that wouldn't have been here before? Also, as an aside, it'd be interesting to know of species that are now extinct. Is there an instance where an animal was returned to Europe from the New World, still exists there, but has become extinct in the US?
I'll discuss New England. William Cronon's Changes in the Land is a classic in the field of environmental history and deals with this region. "The Americas" is a very big zone, and one issue to consider is that of changes in the range of certain species. An animal might not be actually extinct, but its range might have become circumscribed over the past 500 years. Maybe the most famous example would be the American bison, which used to hang out as far east as Kentucky, and now lives only in small pockets in the mountain states and provinces. So looking at a specific region, such as New England, might be helpful.
The earliest European observers of New England saw a number of animals that can no longer be found there. Among them were the gray wolf and the passenger pigeon (extinct). Chestnut trees used to be very common, but were killed off by blight in the early 20th century.
The fur trade caused the local extinction or near-extinction of beaver, moose, otter, muskrat, fox, lynx, marten, raccoon, and mink. By the turn of the 18th century the fur trade had collapsed. Overhunting caused the local extinction of wild turkey, white-tailed deer, elk, and bear. In some cases, these populations saw somewhat of a recovery in the 20th century.
Europeans deliberately brought with them many animals that were not native to America, including cows, pigs, horses, sheep, cats, goats, and honeybees. These domestic animals had a tremendous ecological impact.
Europeans unintentionally brought several microorganisms which were not native to America--chicken pox, influenza, plague, malaria, yellow fever, and tuberculosis. These diseases caused a dramatic depopulation, with mortality rates of 80-90% in New England.
Other important unintentional stowaways were the Hessian fly (which destroyed wheat crops), the black fly, cockroaches, the gray rat, the house mouse, black stem rust (which also wrecked wheat crops), bluegrass, white clover, dandelions, nightshades, stinging nettles, plantain, and many others.
Is there an instance where an animal was returned to Europe from the New World, still exists there, but has become extinct in the US?
I'm not sure what you mean by "returned;" I suspect this isn't the word you meant to use. If you mean an animal that was native to North America and taken to Europe, only to see the American population disappear, I can't think of any examples.