Consequences of $NP=coNP$ and $P\ne NP$?
To me, one of the most basic and surprising consequences of $\mathsf{NP}=\mathsf{coNP}$ is the existence of short proofs for a whole host of problems where it is very difficult to see why they should have short proofs. (This is sort of taking a step back from "What other complexity implications does this collapse have?" to "What are the very basic, down-to-earth reasons this collapse would be surprising?")

For example, if $\mathsf{NP}=\mathsf{coNP}$, then for every graph that is not Hamiltonian, there is a short proof of that fact. Similarly for graphs that are not 3-colorable. Similarly for pairs of graphs that are not isomorphic. Similarly for any propositional tautology. 

In a world where $\mathsf{P} \neq \mathsf{NP} = \mathsf{coNP}$, the difficulty in proving propositional tautologies isn't that some short tautologies have long proofs - because in such a world every tautology has a polynomially short proof - but rather that there is some other reason that we are unable to find those proofs efficiently.