Just some random thoughts off the top of my head...
With most speakers, the thermal power handling limit is the point at which the glue in the voice coil melts and the voice coil deforms. How much wattage can be applied to the speaker before this happens depends on the size and construction of the voice coil, the type of glue, the heat sinking around the voice coil, and the efficiency of forced-air cooling (if any). Note that magnet structure heating can lead to permanent loss of magnetic flux with neodymium and alnico magnets so this can be a limiting factor; with ceramic magnets an overheated magnet will lose flux temporarily but recover its strength once it cools off.
There are several yardsticks used to express thermal power handling. This is from memory, so I'm probably not 100% accurate, but here goes:
There's AES power handling, which is a long-term thermal limit using pink noise as a source.
There's RMS power handling, which is a shorter-term, less conservative limit.
There's music power handling, which is less clear-cut, typically more optimistic, and is based on some sort of "average" music signal.
Then there's peak power handling, which is usually much higher than any of the above. I've seen claims of peak power handling in the kilowatt range for tweeters that couldn't handle 50 watts AES.
And then there is mechanical power handling, which is usually less than the thermal power handling at frequencies below the system's bass resonant frequency.
It's quite possible that a small speaker could have a voice coil and magnet structure with better thermal properties than that used in a larger speaker. It's also quite possible that the manufacturer of a small speaker could be using "peak power handling" or "music power handling", while the manufacturer of the large speaker is using a more conservative yardstick.
Assuming we're comparing apples to apples as far as how the power handling figure is derived, we should also look at efficiency. If the large speaker is 10 dB more efficient than the small speaker, then it will play as loud with 30 watts as the small speaker is playing when it tops out at 300 watts. Louder, actually - by the time we get to maximum rated input, most speakers are thermally compressing by at least 3 dB (and usually more).
Now assuming equal efficiency (and heck why not - equal box size too), the speaker with the higher power handling will probably have less thermal compression. Not always, though - the one speaker may have the larger, slower-heating, better-cooled voice coil but also use a glue that melts at a lower temperature.
Higher power handling isn't necessarily an indication of sound quality either. Many extremely thermally rugged prosound drivers frankly don't sound very good. I've got several such drivers in my basement closet.