The creature takes damage from both spells.
Moonbeam is cast on an area, not a creature. It then causes damage to creatures within that area. The effect of said damage is instantaneous, and so the rules for overlapping spells doesn't apply.
The Melf's meteors example has no relevance here. That's a spell stacking with itself, which is allowed, unlike a spell stacking with another casting of the same spell.
@RyanC.Thompson Ok then, what if two casters each use Melf's Minute Meteors, and attack the same creature on the same turn? Or if you want it to be even more simultaneous, both casters could hold an action for the monster to move forward, and then hit it at the same time. By the logic you state, the monster would only be affected by the more potent of the two meteors.
I don't know why you think moonbeam somehow ignores the rule. The issue is that it's multiple instances of the same spell that would trigger at the same time, and the rule prevents them from stacking. The exact example of bless in the rule similarly triggers "Whenever a target makes an attack roll or a saving throw"; by your logic, you could just say "the bonus to the attack or save is instantaneous, so the rule doesn't apply" to bless... which obviously makes no sense.
I think the problem is that we are now arguing about 2 different questions. I believe the intent of this question was to ask whether 2 instances of the same spell with non-overlapping areas of effect are still subject to the no stacking rule when affecting a single large creature that is in both areas of effect. But this answer is answering whether overlapping damage-over-time effects stack. Unfortunately both questions are relevant to the specific example of moonbeam, which confuses the issue.
I think the important distinction is duration. The +2 bonus from Bless has a duration, and the benefit can overlap. Damage on the other hand doesn't have a duration, and so the benefit from it can't stack. Think about the situation where two druids cast moonbeam on the same creature, but the first one cancels their spell before the other's lands, versus the situation where they don't cancel it. Why would the former result in more damage than the latter?
@V2Blast I don't think that errata actually changes anything. Being ignited by a fire elemental is a condition with a duration. Taking damage from Moonbeam isn't a condition inflicted on someone, but a result of being in (or entering) said space. For example, that same Fire Elemental ability should still deal the initial damage due to proximity.
@Agentpaper, I think it's clear that the DMG version shows that damage is an effect. And that the target takes the most potent one (the one that gives the most damage) only, ignoring others.
@ScottDunnington I disagree. It shows that damage over time effects are an effect, but doesn't address the initial proximity-based damage of the fire elemental. Moonbeam isn't a damage-over-time effect.
@AgentPaper, I think that damage clearly affects so I'm inclined to call it an effect. I don't think the intent was to define effect as only multi-turn damage, but I guess that 's what voting is for. ;^)