"That is our room."
Translation:Das ist unser Raum.
In this case there is no difference. But "Zimmer" is only used when you mentioned the rooms in a flat or building. "Raum" you can also use for this but it can also be used as translation for "area", "sector" and "distance". For example "the distance between us" (der Raum/die Distanz zwischen uns), " at the policitcal sector of Hungary"(im politischen Raum von Ungarn), "in the area of London you find a lot of footbal fans"(im Raum London findet man viele Fußballfans).