Vrăjmaș is an archaism. You won't ever hear it outside of folk tales or historical texts. Meaning is identical but maybe more of a political context
I think I can understand why it is archaic. Hindi also has "dușman" as "enemy". Maybe it's Proto Indo-European.
Probably... In Romanian it has been identified as coming from Turkish in medieval times
Dushman is a persian word, that was borrowed to romanian from turkish. Also I know that hindi borrowed many words from persian.
DUȘMÁN, -Ă, dușmani, -e, s. m. și f., adj. 1. (Persoană) care are o atitudine ostilă, răuvoitoare față de ceva sau de cineva, care urăște ceva sau pe cineva; vrăjmaș. 2. Inamic (1) (în război).
VRĂJMÁȘ, -Ă, vrăjmași, -e, s. m. și f., adj. 1. S. m. și f. Dușman; inamic. 2. Adj. (Despre oameni) Dușmănos, învrăjbit. ♦ Rău, crud, câinos, hain. ♦ (Despre locuri, vreme etc.) Primejdios, potrivnic; cumplit, grozav.
I recommend https://dexonline.ro/definitie/vr%C4%83jma%C8%99 you can find there also examples. Yes, both have the same meaning
Fascinating discussions here. "Enmity" is a very literary word in English, though it certainly does exist. "Hatred" is the normal word, I think. (English native speaker)
Something like that. Dușmănie could be translated as "hostility" or "animosity".