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What's the difference between them? When would you use one over the other?
If possible, please could you explain making reference to these examples, where the usage seems identical.
Example 1:
But the other guests are very noisy! ¡Pero los demás huéspedes son muy ruidosos!
Example 2:
Do you have in other sizes? ¿Tienen en otras tallas?
While "demás" is always used with the definite article (los demás, las demás), "otros" can be used without an article (meaning "other(s)") or with an article (meaning "the other(s)").
Thus, your example 1:
means that there are no other guests left to be rated as quiet or noisy. The speaker and his group are quiet, while all the other guests are noisy.
In example 2, reference is being made to other sizes than the one the customer has been shown. There may still be some others the sales assistant does not have available. "otros/otras" is therefore non-restrictive, or indefinite. Once it has been used, the definite article may appear.
Supplementary answer to add to what @Gustavson wrote:
You can think of demás as the remaining ones, the remainder, or the rest. Let's take your example first:
¡Pero los demás huéspedes son muy ruidosos!
But the remaining guests are very noisy!
or
But the rest of the guests are very noisy!
Here's another example. Let's say you're helping your younger brother sort through the clothes strewn on his bedroom floor.
Tú: ¿Esta camisa está sucia?
Tu hermano: Creo que sí.
Tú: ¿Y las demás?
Here's an English version of the conversation:
You: Is this shirt dirty?
Your brother: I think so.
You: What about the rest [of them] / the remaining ones?
This means, the remainder of shirts still strewn on the floor.
Now a conversation with otras. You're preparing to do a load of laundry with your brother.
Tú: Todavía hay espacio en la lavadora. ¿Tienes otras camisas sucias?
Tu hermano: No, ya te di todas.
In English:
You: There's still room in the washer. Do you have any other shirts needing washing?
Your brother: No, I already gave you all of them [my dirty shirts].
One more tip. If your expression could include some or any, then you'll probably want to use otro/otra/otros/otras.
You can think of "otros" as non-exhaustive and "demás" as exhaustive. I usually pick "demás" in cases where English says "everyone else" or "everything else."
