Starfish Prime's answer addresses what happens if a full lunar mass falls onto an Earth like planet. I'd like to reality-check the event itself.
In order for the Moon's mass as a whole to fall on the planet, you have to deorbit it. The amount of energy you would need to do so lies within the range of the kinetic energy you get from the impact, give or take one order of magnitude.
For the Earth this would mean resurfacing and remodeling but for our own Moon, which is two orders of magnitude less massive, this could mean total obliteration. A quick search on the internet and I found out that the gravitational binding energy of our own Moon is around 1.2 $\times$ 1029 joules - 20% of the energy you'd get from the final impact on Starfish Prime's answer. I understand that the falling moon in the question would be less massive so it would have even less binding energy.
So... The moon will become quickly expanding plasma way before its mass deorbits onto the host planet.
Much of this plasma will probably be made of particles going much over the escape velocity at the moon's altitude, and the bulk of the remaining mass would just settle in new orbits around the planet as rings. The bigger moon may get somewhat scorched, but the planet might keep its atmosphere mostly intact. Over weeks or months that plasma would cool down and become ionized gas, which would deorbit over eons so no danger there.
Edit: Starfish Prime put one some calculations here. A lot of that plasma will hit the planet - around 10 dino-killing asteroids worthy of plasma, considering the upper margin estimates for the asteroids mass. Though the impact would be spread across the side of the planet facing the plasma, the effect would be much the same. So... Mega extinction event. Life might still exist in the oceans, but the surface would probably be [redacted].
This also means a large mass that was astronomically close the planet has been removed from the system. There might be some earthquakes as the crust of the planet readjusts. Moons and planets stretch each other like a drawn bow; As the mass of the moon leaves the system the planet will contract like the same bow being slowly undrawn. That is true even if the bigger moon stays.