Yunhee J. answered 04/19/21
Experienced Korean Language Tutor 10+yr private lessons, Korean School
원 is Chinese oriented word. 동그라미 is pure Korean.
They both mean "circle "
There is no contextual requirement when to say 원 and 동그라미.
However, sometimes when you write, you'd like to put the similar sound words together.
There is a verb 원하다 meaning "want"
동그라미, the sound is a bit poetic and pretty to the native ears.
동그랗다 = to be round (feeling or implying "tiny", "small", "cute"... etc ) 동글동글 adv.
둥그렇다 =to be round (feeling or implying "bigger" circle shape) 둥글둥글 adv.
There is a verb "물끄러미" meaning silently gazing at something
I would match 원 and 원하다, 동그라미 and 물끄러미 when writing. ^ ^
But this is just poetic, rhyming purpose and there's no difference in meaning of the two words.