~와/과 means 'together with'.
오늘은 어제와 다르는 것 같아요 (today seems different (together) with yesterday)
A와 B (A and B, A together with B)
Therefore 옛날과 달랐어요 means 'it was different with before (old days)'
The subject was omitted, the full sentence could be :
그땐 옛날과 달랐어요 (At this time it was different with the old days)
But it depends on the context you've read this phrase.