Does the wattage on a power supply simply mean the max output wattage?
The wattage implies the maximum available wattage to the system.
However note that the PSU draws AC power from the wall socket, converts it to some other DC voltages, and provides those to your system. There is some loss during this conversion. How much depends on the quality of your PSU and on how much power you draw from it.
Almost any PSU is very inefficient when you draw less then 20% of max rated power from it.
Almost any PSU has less than peak efficiency when you draw close to the max rated power from it.
Almost any PSU has their optimum efficiency around 40% to 60% of maximum load.
Thus if you get a PSU which is 'just large enough' or 'way to big' it is likely to be less efficient.
[But note that your PC does not consume a fixed or constant level of power. At idle, when not much is happening, the DC power consumed will be low. Perform a lot of processing and I/O operations, then power demand goes high.]
Bellow is a nice example of a real world efficiency graph for 900W PSU. The x-axis shows the power supply load in Watts and the y-axis shows efficiency.
will there be any tangible difference in the outlet wattage draw between a 1200W power supply vs, say, a a 800W power supply?
The 800 Watt PSU would run at 62.5% of max rating. That is a good value.
The 1200 Watt PSU would run at only 41% of its maximum rating. That is still within the normally accepted range, but at the low end. If your system is not going to change than the 800 Watt PSU is the better choice.
Note that even with a good (bronze+ or silver rated PSU) you are still loosing about 15% during conversion. 15% of 500 Watt means that your computer would use 500 Watt, but the PSU would draw 588 Watt from the wall socket.
I am aware this is not the correct answer, but cannot delete it because it is accepted. The right answer is given by Hennes.
The wattage of your power supply is what it could potentially supply. However, in practice the supply won't ever make that. I always count 60% of the capacity as the truly maximum capacity. Today however, there are also bronze, silver, gold, platinum power supplies which guarantee a certain amount (minimum of 80%) of efficiency. See this link for a summary of 80 PLUS labels.
Example: If your 1200W supply has a 80 PLUS label on it, it will supply probably 1200W but will consume 1500W. I think you 800W supply will be sufficient, but it won't guarantee you safety.