How would the fight at the end of Captain America: Civil War been different if Bucky had his now vibranium-infused metal arm?
It would have gone the same way, right up to that crucial moment where Tony's chest beam disintegrated Bucky's arm. At the time, Bucky was about to rip out the suit's Arc Reactor, which would have disabled Tony (the same way Cap eventually did).
With a Vibranium arm, the beam would have probably hit the palm and then either deflected elsewhere (around the fingers/palm) or bounced back at the suit. Either way, Bucky keeps going for the Reactor and either tears it out or destroys it via reflected chest beam.
Or it explodes against the palm in a similar manner as the scene in Iron Man 2 where Rhodey and Tony hit each others' repulsor blasts, knocking both Bucky and Tony out.
Either way, the fight ends the same way.