How diverse are species in remote, isolated areas on Earth?
The concept that you hint at, has been studied at the example of the island of Australia, where marsupials often look quite similar as their counterparts in the same ecological niche, which live outside Australia and have a placenta, and are less related on the DNA level (e.g.: squirrel-like animals, wolf-like, and the now everywhere extinct saber-toothed tiger-like...).

If you are interested to learn more, I would highly recommend to read Jerry Coyne's "Why evolution is True" ( http://jerrycoyne.uchicago.edu/index.html )

The specific example of Europe and Americas (which does not fully capture the complexity of your question), also is quite interesting - although it is very short on an evolutionary time-scale. E.g.: American squirrels, which lack predators in Europe, and only arrived in Europe because of human journeys following the rediscovery of Americas, are partially outcompeting the beloved red squirrels ( https://en.wikipedia.org/wiki/Eastern_grey_squirrels_in_Europe )