사이가 안 좋아요 and 관계가 안 좋아요 have about the same meaning. Generally, 사이 is more colloquial and suitable for common relationships between people or other entities. 관계 is more formal, and it often represents more complex or subtle relationships. So an adult might tell two boys 사이좋게 놀아라, but 관계 좋게 놀아라 would sound ridiculously out of place. 사이 is also quite limited in possible expressions, mostly evolving around 사이가 좋다, 나쁘다, 좋아지다, 나빠지다, etc. whereas 관계 has a wider variety. 사이 has another major meaning of a gap or separation between two objects (두 도시 사이의 거리). And 관계 has the important meaning of a logical/conceptual connection, as in 이 일은 너와 관계없다 (This has nothing to do with you), which can't be expressed with 사이.

That is exactly same meaning :)