Door, gate, portal, postern, doorway, gateway are comparable chiefly as meaning an entrance to a place.
Door applies chiefly to the movable and usually swinging barrier which is set in the opening which serves as an entrance to a building or to a room or apartment in a building.
Sometimes door is used also of the opening.
Gate may apply to an opening in a wall, fence, or enclosure but it more commonly denotes a movable and often swinging barrier (especially one made of a grating or open frame or a heavy or rough structure) set in such an opening and closed or opened at will.
Portal applies usually to an elaborate and stately door or gate, with its surrounding framework.
Postern denotes a private or retired door or gate (as at the back of a castle or fortress).
Doorway and gateway apply not to the structure but to the passage when a door (in a doorway) or a gate (in a gateway) is opened for ingress or egress.
In their extended use these words are still more sharply distinguished. Door usually applies to what provides opportunity to enter or withdraw or makes possible an entrance or exit.
Gate differs from door chiefly in its connotations of facility in admission or of entrance into something large, impressive, wide, or even infinite.
Portal often carries similar connotations, but it usually applies to a definite place or thing which is itself splendid or magnificent and through which something (as the sun at rising and at setting) is admitted or allowed exit.
Postern, on the other hand, implies an inconspicuous or even a hidden means of entrance or escape.
Gateway is usually preferred to doorway in figurative use because it more strongly suggests a passage through which entrance is gained to something desirable or difficult.