Here are the key differences between the pairs of sentences:
Pair 1:
S1 uses an idiom - "raining cats and dogs" which is an informal way to say it is raining very heavily.
S2 directly states it was raining very heavily in a more formal, literal way.
Pair 2:
S3 uses an active voice construction - "We believe..." which indicates the subject "we" has the belief.
S4 uses a passive voice construction - "It is believed..." which removes the subject and makes the belief more generally attributed.