There is actually a pion exchange between nucleons. This is the basis for the residual strong force.
When a proton and neutron interact, one of the interactions we get is
$$\rm proton^+ +neutron \rightarrow proton^++pion^-+proton^+ \rightarrow neutron+proton^+$$
So the proton becomes a neutron and then the neutron becomes a proton by transferring a negative pion.
Edits: <PERSON>: That's fair, my answer didn't really answer.
The proton and neutron aren't identical particles and so aren't limited by Pauli Exclusion between each other. They can have the same spin state and for some reason having parallel spins gives the combination a lower energy and so more stability.
When I look for why parallel spins have a lower energy, I just see it stated as a fact or in better references, an equation that gives the details of the energy difference is given.
What I haven't seen is a physical explanation for why parallel nuclear spins have lower energy.
Here's my hunch (which I will try to verify). If it's correct then it's probably out there somewhere already. 969
If it's wrong, it's because I just thought the idea through myself.
When a proton and neutron have all the same quantum numbers (and so their spins are parallel), then when a neutron transfers a negative pion to the proton, the final state is identical to the initial state. This means there are two different paths from the initial state to the final state: doing nothing and passing a negative pion both give you the same final state.
In quantum when two different events are possible because they both give the same initial and final state there is an interference term. For instance, in the collision of two identical atoms with identical spins states, colliding and not colliding can give the same final state, and so the two possibilities interfere.
In the case of a proton and a neutron with antiparallel spins, transferring a negative pion would give a different final state than the initial state. It could transfer back to then give the same final state, but (roughly speaking) every step a process needs makes the contribution of that process less important.
In the case of a diproton or dineutron, the transferring a neutral pion would also give the same final and initial state.
So I further suspect (again, all on me if it's wrong) that while the proton-neutron parallel pair has dominant exchanges of both neutral pions negative pions, the diproton and dineutron only has neutral pions giving the proton-neutron pair about twice the binding.
So the two key ideas I'll try to track down are: (1) Does a process having identical initial and final state make it more important? (2) Does having twice the number of similar interactions mean that there is about twice the binding energy involved? 969