"희망" is a 한자 (Sino-Korean word), whereas "바라다" is a pure Korean verb. Thus, "희망하다" is more formal and written than "바라다". You are supposed to say 바라다 more often in your spoken Korean.

They are very similar but 희망하다 sounds a lot more poetic and formal. 바라다 also sounds poetic but is used pretty often. Like, 여러분의 밝은 내일을 희망합니다(already sounds like an advertisement copy) 네가 내일은 아프지 않기를 바라. 올해는 애인이 생기기를 바라고 있어요. 좋은 하루 보내시기를 바라요. But 희망해/희망해요 would be a bit awkward for the first two sentences.

Thanks 😊