Manufacturer speaker power recommendations

Using the higher power amp is less likely to damage the tweeter, provided of course that you resist the urge to turn it up to 11, which is more likely to occur if you use a lower power amp. Higher power suggested limit is most likely, IMO, to protect manufacturer from someone using a 1KW amp and then asking for warranty replacement when they turn it up to 11.

It is more complex than a single number. A few badly distorted watts are worse than a moderate number of clean undistorted watts but as Swampwalker said, not a zillion clean watts.

"Using ie. 300/400w amp doesn't damage speaker, does it?"

It depends. It doesn't really matter how powerful the amp is as long you keep the volume to whatever your speakers can handle. Just use common sense when adjusting volume. If you hear something straining, turn it down.

Maybe Ralph from Atmasphere will pop in on his thread;if he does pay attention to his response.

The recommendation by a spkr manufacturer is for minimum watts needed to drive the spkr and for nominal wattage, not peaks. Peak wattage is usually 4x higher, but that is a very short duration.
A 300 watt amp would mean it could provide a peak signal easily without clipping. You probably would only be using about 10 to 15 watts for typical listening.