Try using one equal sign:
[Opportunity].Record_Type_Name__c = 'New License' && [Opportunity].StageName = 'Closed Won - Not Paid' && NOT(CONTAINS([Opportunity].Type, 'Additional'))
EDIT: Found this note in the guide...
There must be a space between the function’s parentheses and the
contents of the parentheses. For example, you’ll get an error for
TEXT([Account].LastModifiedDate) but not for TEXT(
EDIT2: Found an article that states picklist formulas are not supported and in the above guide there is a generic picklist not supported everywhere.
Either tool will work however workflow may be the better option right
now. There is no function help in process builder so complicated
formulas may be difficult to build out. Also picklist formulas do not
work in process builder. - See more at:
Processes evaluate picklist fields as if they’re text fields, so some
functions that normally support picklists may not support those