Does the order of columns in a columnstore index matter? I have a table with ~200 million rows and ~15 columns in it. I am planning to create a COLUMNSTORE index on my table. 

Will there be any change in performance based on the order of columns that I use in the columnstore index? If yes, what is the logic behind it?
No, order does not matter. Each column is considered individually.
From the SQL Server team (emphasis added):

Typically you will want to put all the columns in your table into the columnstore index.  It does not matter what order you list the columns because a columnstore index does not have a key like a B-tree index does.  Internally, the data will be re-ordered automatically to get the best compression.