It's difficult for me to tell you what to do from here, I'm in Australia and we may have different laws etc.
Generally, the person on the low side of the retaining wall is responsible for it.
Generally, for walls over 1m high they need to be council approved and engineer spec'd.
Generally, a person (like your neighbour) needs to be advised/forewarned of the impending problem hopefully prior to damage ... however, that does not in itself negate a contribution if in fact activities on his side of the wall contributed to it's failure. So it gets sticky. And ugly, and as usual the tree owners think it aint them and a fault of your wall.
But ask this question, if the trees (or palms) weren't there would the failure have occured?
Also ask this question, if the trees/palms were planted after the wall was built then why would the wall have had to have been engineered for the force put upon them by the trees?
In other words, the wall wasn't engineered to take the force of the trees.
I have no doubt those palms added to the failure of that wall. Some will argue, they like to argue rather than open their wallets I find. They might suggest a section (worst section where cracked) is opened up for a peek inside and if there's no roots pushing against it you have no case (so they think). However what can happen is as the root mass expands so does the soil around it, in turn adding pressure to the wall.
Now, what I have done is sketched a basic diagram to explain some events.
See, what happens is now, like a sailing ship has a mast, there's a pole with sail creating leveraged soil movement. Also, the tree doesn't want to starve or fail so it creates more roots.
The environment under that concrete driveway isn't as good as the wall side so a greater concentration of roots will be wall side.
Bigger root mass, leverage against wall as your soil level isn't there to offer resistance and WHELLA, a bowed out wall and concentration of cracks right where the palm is.
Now lets go back a little and say you pulled a few bricks out but there's no roots to nail the neighbour.
Doesn't have to be, you see, like a plant root bound in a pot the force inside the neighbours side exceeds the force on your side. The roots might not be touching the wall but they have expanded soil volume which is in contact with your wall.
From experience I could comfortably say that if you did knock out a few bricks you'd find a mass of roots. Dont forget that in most cases water will run off to the wall then go downwards ... palms will home in on that.
It's easy for people to plead ignorance, easy for them to say it was a design flaw. It's just as easy for you to say that if the trees weren't there it wouldn't have happened.
I think, in all fairness and equality of the matter, based on my knowledge and experince, that a 50/50 resolution to repair would be fair providing that the vegetation was removed or in the event of the neighbour wishing to retain the vegetation he pay the increased costs of a wall engineered to take the additional maturity force of those palms.
You want trees, you pay for them not the victim.
I also have retaining walls on my property. My neighbour, who I explicitly advised recently to either remove (free done by me) or reduction prune a melaleuca on his side near my wall to prevent wall breakage has mysteriously crown lifted it. This is the worst thing he could have done and directly opposite of the advice I gave, it promotes apical growth and provides more of that leveraged wind force and will promote more growth and an expanding root system.
When I questioned him about this he replied, "he'll be long dead before it's a problem." Not funny if he moves and the new owner says they like the tree and it stays ... yada yada yada wall cracks and the tree huggers reckon it's my fault. Our wall is right on the boundary like yours, he is higher by 3' (worse than yours) and I'm an arborist having to deal with yet another PITA. I believe any reasonable person would have obliged, kept the piece, pruned or pissed it off and we all live happily ever after ... sadly, this aint a fairytail like your event.
You could always give him the link to this thread to get the ball rolling or start with getting an arborist and engineer and evidence.
I wish you well, and they are just crap ugly palms anyway!