I think you can not do the first method because the vector is a container that contains string which is also a container... You can only do your first example between vectors.
You can do it by your second but you can not declare the size of scr[] array like that. You have to do it at runtime. So you have to use pointer to string.
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vector<string> screen;
...//add data to screen here
string *scr;
scr = new string[screen.size()];
for(int i=0; i<screen.size(); i++){
scr[i] = screen[i];//Copy the vector to the string
}