Adele's song says: "It burned while I cried". For me, I learned that I should use past continuous after while or past simple after when. so the sentence should be eighter "It burned while I was crying" or "It burned when I cried". The sentence with when doesn't really make sense for me so I think they should use the sentence with while but with the past continuous tense. What's the difference in meaning between the sentence in the song and my sentence?
There's nothing wrong with using past tense like this. It is common and correct to say
I played with my phone while he watched TV.
There is no grammatical requirement to use a past continuous with while.
Using "past while past" suggests that the two activities occurred at the same time whereas "past while past-continuous" would suggest that the first event started and finished during the time of the other activity.