I know that разве has some additional meanings, for instance the expression разве что/лишь/только. However in their principal meaning, разве and неужели are somewhat synonymous.
What are the nuances of difference between the two words? Are there some situations where you would definitely use one but not the other?
Неужели is particle used in questions only. On the other hand, разве can be both particle and adjunction in both questions and positive statements.
Within a statement разве is often used together with other words such as что/лишь/только/иногда/когда etc.etc. But the latter is not a strict rule. For example, "Смолкли осенние птички, разве изредка вдали дергач прокричит, сова ребёнком заплачет, филин ухнет в бору". Here "разве что" is intentionally changed to just "разве" without losing sense.
Now considering question sentences, here the situation is more controversial. Разве and неужели seem to be almost synonyms to each other. But, I think, there's a little nuance - IMO, разве is more neutral and commonly spoken word, while неужели is more "pushing" one. For example:
Разве ты не придёшь? --> Just surprised.
Неужели ты не придёшь? --> Some note of disappointment; maybe you'll change your mind?
"Неужели X?" means that X is possible but surprising. The surprise may be pleasant or unpleasant. Example:
Неужели вместе нам
Песню не сложить?
Неужели суждено
Hам в разлуке жить?
(From the song "Неужели", lyrics by И. Шумаков)
The singer doesn't know whether "суждено нам в разлуке жить" or not.
"Неужели" can also refer to good things, i. e. "Неужели Маша меня любит?" or "Неужели суд меня оправдает?", which means that the speaker considers it probable, albeit surprising.
"Неужели?" or "Да неужели?" can be a complete sentence. It means "Really?" and expresses the surprise (but not disbelief) about what you just heard.
"Разве X?" means that X is improbable, if not outright impossible. If one told instead "Разве суждено нам в разлуке жить?", it would mean that he doesn't believe that "суждено нам в разлуке жить".
"Разве" is often used with rethorical questions. Example from Poroshenko's speech:
Но поставлю и себе, и вам риторический вопрос: разве Россия когда-нибудь выдаст нам преступника Януковича, главного виновника кровопролития на Майдане?
The word "разве" means that, according to Poroshenko, it is obvious that this will never happen.
Just like "разве X?" means "not X", "разве не X?" means X. Example:
Разве не я тогда у креста,
Разве я не тонула в море,
Разве забыли мои уста
Вкус твой, горе!
(Anna Akhmatova)
The common idioms with "разве" are:
