It is very interesting, given that the US is directly involved in Syria, whereas Denmark is not.
It has much to do with how people integrated into Europe v America.
Europe is very difficult to integrated into - they are ethnic groups. You can mostly 'become American' but it's almost impossible to 'become Danish' unless you are there from a very early age.
Americans in particular have a hard time with this, because they don't 'see' ethnicity, and often use the term 'race' which is the wrong term to describe groups. 'Race' is genetics, not particularly meaningful. Ethnicity is a set of values, behaviours, cultural norms, i.e. attitudes towards, family, government, education, religion, etc. etc..
Most nations outside the Americas are simply formalized governments of ethnic groups.
An American passport pretty much makes you American. A Danish passport makes you a Danish citizen, but not necessarily Danish.
Also - Europeans are more intellectually open minded, but more behaviourally cloistered, whereas Americans are sometimes a little less intellectually open (i.e. popular ideas, policies), but in practical reality, almost always more 'open'. Euros talk benevolence but hang out with their own kind. Americans sometimes talk anti-immigration, but they are pretty fluid in terms of social and business relationships.