Both the words next and after have similar meaning, but their usage is different. next means immediately following something but after can mean any time afterward without caring about the order.
Both next and after are preposition. Whereas next is an intransitive preposition, after is transitive except in cases where it has an implied elliptical complement. next can also be an adjective.
After lunch we left for a movie. [Preposition after takes a complement lunch]
Look that is the thief running, and the police chasing after. [Here the preposition after doesn't have a complement, but it has an implied elliptical complement like this *...and the police chasing after him (the thief)]
But we the preposition next is an intransitive, that is it doesn't allow any complement.
- We left for a movie next lunch. [INCORRECT]
What are planning to do next? [CORRECT]
In constructions like next day, next Monday it's an adjective. It's actually an NP that can act like an adverbial. Lerson called it bare NP adverbial.
We will be back to school next Monday.
We will be back to school after Monday.
Though the sentences above syntactically look similar they mean different. While the one with after means that we can be back to school any day that comes Monday. But the last sentence says that we will be back to school on Monday that comes next.
Hope this helps. I wish I could reword the last paragraph, but I can't think of any other way to write it right now.
He is in the middle of a task.
He is in.
He is subsequently [INCORRECT]
But we can write the following -
Our performance is next.
- There are certain adverbs (e.g right, straight) that can only modify a preposition.
Stand right next to me.