I believe the differences between conservatism as it is known in Europe, Canada and Australia on the one hand and how it is known in the United States on the other hand needs to be investigated and properly understood.
The Republican party has developed a very special brand (or brands) of 'conservatism' that is (are) drifting further and further away from mainstream conservatism.
While I'm not, as my name suggests, a conservative I have some respect for mainstream conservatism as a counterbalancing force in political life and I rarely - if ever - doubt the democratic sincerity of mainstream conservatives.
But since I see more sinister forms of 'conservatism' in the Republican party I would like to highlight a few areas where European conservatives would find precious little common ground with Republican 'conservatives'.
(My perspective is shaped by the fact that I am Swedish but I have spent plenty of time in both continental Europe and the U.S.)
* Abortion. While there are some exceptions (notably Catholic nations like Poland and Ireland), a woman's right to choose is settled among European conservatives. By most it's not even seen as a political issue but as a private matter. A Swedish conservative who tried to turn abortion into a political debating point would find very little support in his own party. Likewise, it's uncontroversial that public schools should teach children about (mutually respectful) sex and contraceptives.
* Gun laws. While Sweden is a nation of hunters, especially in the North and in the countryside (moose is probably the most preferred prey), gun access is heavily restricted. There is no debate about gun control because Swedish conservatives agree with this policy which is tailored after the needs of hunters, collectors and sports shooters but makes it very difficult for anyone else to have access to guns.
* Government and taxes. To be anti-government wouldn't be considered a conservative position in Sweden (or Europe), it would be called anarchism. While many conservatives in Sweden would like to shrink government and reduce taxes, government is not seen as evil. I dare say that most conservatives see government as a constructive force in dealing with collective problems. The present centre-right-government in Sweden has mainly (with some exceptions) reduced taxes in a way that benefit the working poor.
* Government and religion. Swedish conservatives are not comfortable with religion mixing into government, you wouldn't even here a Swedish conservative utter the words "God bless Sweden" (they would be seen as strange and sectarian, why on earth would God bless a nation?) This is true in most of Europe, when Tony Blair wanted to end a speech with "God bless Britain" his staffers protested and stopped him.
* Science. Any conservative politician who denied evolution would be seen as an extremist and wouldn't be electable to any kind of office. He would also be ridiculed within his own party. While there is certainly a fringe debate about the science behind manmade climate change, mainstream conservatives are onboard with the conclusions of the IPCC. A mainstream conservative in Sweden wouldn't oppose reigning in CO2-emissions, the pace and the methods of course are debated but most conservatives no longer oppose cap-and-trade or carbon-taxes.
* Capital punishment. (I am aware that some Democrats support the capital punishment.) A nation which hasn't abolished capital punishment can't join the European Union, that debate was settled a long time ago. A conservative European politician who advocated capital punishment today would be seen as extreme.
* Health care. A European conservative wouldn't question that universal healthcare is a right of every citizen, rich or poor.
More examples could be added (for instance the United Nations, foreign policy in the Middle East, gold buggers, civil unions or same-sex marriage etc.) but my point is that mainstream conservatism in Europe and Republican 'conservatism' don't share a lot of common ground and they are drifting further apart.