okay
‘괜히’는 ‘괜히 했네’, ‘괜히 왔어’ 이런 예시 중에 쓰이는 뜻을 알겠는데, 아래의 문장에서 ‘괜히’를 쓸 때 무슨 뜻인지 조금 이해가 안되네요!! 그리고, 이런 상황에서 뭔가 ‘왠지’를 써도 되는데요....혹시 두 단어의 차이가 뭐에요??

요즘은 괜히 짜증이 난다.
요즘은 왠지 짜증이 난다

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There is almost no difference with the meaning, but the difference is in the nuance.
"괜히" gives a negative feeling. Like a regretful feeling. "왠지" gives the feeling exactly the same as the meaning of the dictionary.

Let's look at the sentences you gived.
요즘은 왠지 짜증이 난다. Talking, but the eyes are on the passing clouds. Or talking with
an expressionless face. Giving the feeling that "yeah... Just that."

In the other hand, 요즘은 괜히 짜증이 난다. Talking with a depressed face. And giving more focus on the irritation and stress that he is feeling.

P.S.
My explanation is just a example.
If the speaker is using "괜히", the sentence will mean as I explained.
But the speaker is using "왠지", the meaning of the sentence will be different by the situation.
When the speaker is using "괜히" and "왠지" for a negative meaning, "괜히" will be a stronger expression than "왠지"