Not necessarily. The original scientific term 'Caucasian' , which is no longer used, actually covered residents of southern Asia and North Africa, people who would not usually be considered 'white' (whatever that means). Nobody is really 'white' in colour, or really 'black', for that matter, and it is very easy to cause offence when referring to skin colour or when classifying people according to race or ethnicity. It is an extremely sensitive issue, and is best avoided.

Only in America they are

This has been one of my pet peeves. 'Caucasian' implies that there is some descendance from people from Caucasus and would include many languages including the Indo-European languages. However, in Europe there are two nations/languages that undoubtedly are "white" (not that anyone is white, but it would be rather hard to find anyone whiter than these): Finnish and Estonian. Finnish and Estonian languages and people belong to a totally separate language group. Thus, they are not Caucasian as they are not from Caucasus :D And for sure are not Anglo nor Anglo-Saxon :D

The term Caucasian was used here in the US until maybe the 1980s to describe white people. I remember well checking a box on forms stating that I was Caucasian.

In American English, yes they are the same. You can also say Anglo or Anglo-Saxon