떠오르다 is something surfacing on its own — a memory or thought arriving unbidden (그날의 공기까지 선명하게 떠오르니까요, even the air of that day comes back vividly; 나도 모르게 떠오른, surfacing without my meaning it), or a person rising into prominence (재무장관 후보로 유력하게 떠오르는 인물). -게 gives the manner. If you are deliberately calling something to mind, the verb is 떠올리다.