RTOW: Regulated take-off weight. Effectively the maximum weight at which the aeroplane may take-off with regard to actual conditions of surface wind, atmospheric pressure, runway length, outside air temperature, etc, etc. NOT the same as the maximum permitted stuctural take-off weight. In fact, in some circumstances it is possible for theoretical RTOW to be in excess of max certificated structural take-off weight; this does NOT mean that take-off weights in excess of the certificated limit are permitted on such occasions. The aircraft could probably take-off without difficulty as performance would be adequate, but the aircraft would have been overstressed.
ATOW: Actual take-off weight. For example, some aeroplanes may be permitted to adopt a reduced thrust take-off technique if there is more than a minimum defined difference between ATOW and RTOW - even if RTOW is above certificated structural limits as described above.
Fairly obviously ATOW must never be greater than RTOW!!
For example, if max certified limit for an aeroplane is 185T, RTOW at the time is 186T and ATOW is 185T then fine. But if RTOW drops to 183T because of a change in conditions, take-off must not be attempted!!
Taking off just above max. certificated take-off weight might overstress the aircraft and lead to the aircrfat being grounded for fatigue investigation; taking off above RTOW will probably kill everyone if there is a failure when the aircraft is in gap between being able to stop within the distance remaining or being able to accelerate to take-off and clear all ostacles by the stipulated margin. That is what scheduled performance is designed to prevent; with modern aircraft (especially twin jets with capable brakes) it is often possible to reject a take-off right up to rotate speed - this is certainly NOT the case in older aircraft which, having reached V1, MUST continue the take-off if a malfunction subsequently occurs.
OK pedants, this is a simplification I know, but it's supposed to explain things in lay terms.
[ 22 August 2001: Message edited by: BEagle ]