In general, you split up the WHERE condition at every top-level AND. Then you do either pullup or pushdown or both (it depends a bit on how your planner looks on the inside). In the end, you end up with the two cases being exactly the same before you start planning.

For outer joins (left/right/full), it's different, and there you absolutely need the explicit join syntax (for correctness). And semijoins are not expressed the same way at all, partially for weird historical reasons.