What's the difference between a thousand 0.2W LED's and two-hundred 1W LED's?
Grow lights are about optimizing running costs.  You want a cheap device that uses as little power as possible.  For this reason, very low power LEDs are impractical because you need too many diodes to get usable quantities of light.  Very high power diodes are also less cost effective because they concentrate heat into a small area, reducing efficiency and requiring more costly cooling.  Thus you tend to end up with a lot of mid-powered LEDs.
As for your specific questions:

What's the difference in the light output over a given area, in terms of PAR, PPFD, PPF?

You would have to specify a product before anyone could answer that.

I grow plants indoors under LED panels. People in this industry say to use +1W diodes and that below 0.5W is useless or inefficient. I want to know if there really is any difference between the two.

One of the most widely used and also the most efficient white LEDs you can get in terms of umol/j are the Samsung 301b/h, and those are 0.2W class diodes, so I think that whoever said that is uniformed.

Does the bigger wattage diode penetrate deeper

No.  Higher wattage diodes use more electrical power.  That is all wattage means.  Since different sources have different quantum efficiencies, you should be looking at actual mol/s of photons, not how much power is being consumed (or wasted).
In addition, if someone is talking about &quot;1W&quot; or &quot;3W&quot; diodes rather than light output, they're probably talking about cheap crap you get on ebay with no datasheets and exaggerated specifications.