There are differences between classes and the roles that you can have in a party. In 5e the different classes are very flexible, many classes could fulfill different roles. So usually it´s not needed to have 6 different characters to take care of 6 different roles in the party.
I don't know 5e well enough at all, so I can't comment oh who is right. But I can say I do hope classes mean more. It's ironic, though, since I used to bemoan the rigidity of 'classes' for so long. But maybe because there seems to have been so little of the old school class system recently I am actually looking forward to more of it (e.g no other class can match a rogue in lockpicking).