拝む | Literally, to pray (at a Shinto temple). Figuratively, to worship the ground someone walks on. 拝む also means “to contemplate”, like a stalker his idol of choice, his face like he’s looking at God himself |
祈る | Literally, to pray (Christian style). Figuratively, to wish (for good weather, long life to your friends, to pass this fuckin’ math test). Emphasizes the ACT of prayer, rather than the contents. |
祈願 | A figurative prayer or wish, made to pass a test or to not go to jail, etc. |