She cried before she fell asleep.
She cried before falling asleep.
In English, both are fine, though the second is probably preferable because it avoids unnecessary repetition.
In French, the difference in the way the two sentences sound is more extreme. The first version is unduly "heavy" and you are expected to use the second (especially on a grammar exercise!): we're generally taught to use an infinite construction when the subject of both clauses is the same.