From dictionary.com:
gift: something given voluntarily without payment in return, as to show favor toward someone, honor an occasion, or make a gesture of assistance; present.
synonyms: donation, contribution, offering, benefaction, endowment, bounty, boon, largess, alms, gratuity, tip, premium, allowance, subsidy, bequest, legacy, inheritance, dowry.
reward: something given or received in return or recompense for service, merit, hardship, etc.
synonyms: desert, pay, remuneration; requital; bounty, premium, bonus.
I don't see them as synonyms at all.
http://thesaurus.com/browse/gift
gift
Part of Speech: noun
Definition: something given freely, for no recompense
Synonyms: allowance, alms, award, benefaction, benefit, bequest, bestowal, bonus, boon, bounty, charity, contribution, courtesy, dispensation, donation, endowment, fairing, favor, giveaway, goodie, grant, gratuity, hand, hand-me-down, handout, honorarium, lagniappe, largesse, legacy, libation, oblation, offering, offertory, philanthropy, pittance, premium, present, presentation, provision, ration, relief, remembrance, remittance, reward, souvenir, subscription, subsidy, tip, token, tribute, write-off
Antonyms: forfeit, loss, penalty