There are two very common ways in Spanish to say the equivalent of "again":
- de nuevo
- otra vez
But I use them pretty randomly because I've never been able to pick up on any differences in how native speakers use them. Are there any differences?
Para salir de dudas, recurrí al Servicio de Consultas de la RAE y transcribo aquí la respuesta que recibí:
Así que, las dos expresiones son completamente equivalentes, salvo en algunos contextos muy específicos, como el del ejemplo que aparece en el texto citado.
They are used pretty much interchangeably. Check this page, "Saying again in Spanish", it has some examples.
There are some expressions in English where you would use "again", but that in Spanish change form or that are slightly different:
I would say that "de nuevo" or "nuevamente" sounds more like "all over again", while "otra vez" is just "again".
As noted, these differences are very subtle, in many cases "de nuevo", "nuevamente" and "otra vez" can be used interchangeably. However in some cases "de nuevo" ("nuevamente") is used in meaning "anew", then changing it for "otra vez" changes the meaning of the sentence slightly.