Besides the language being Latin vs local languages, there is one huge difference people don't know about. The Tridentine Mass has the priest facing toward the altar and the tabernacle, this is called "ad orientem". In "modern" day post-Second-Vatican-Council mass, the priest typically speaks the local language and faces the congregation.
The modern mass can be done ad orientem. The switch of orientation was only permitted, not required, in the Novus Ordo.
Most of the changes people associate with the Novus Ordo were completely optional and often not even expected to have become so common. This is why all the popes have been exasperated, to varying extents, with the Latin Mass movement. Literally nothing prevents dioceses from celebrating the Norvus Ordo mass in Latin, ad orientem, chanting, etc.
That's it, I wasn't sure if I was imagining the "priest not looking at you" thing...