Does 600mw video transmitter not give you more range than 200mW? 
I understand how it influence / affect other racers but what about distance? Is it much better?

Yes more power with video transmitter, the more range you get. Also the penetration ability is also better.

My understanding is that in theory, the power and range isn't proportional.

You need 4 times the power to get 2 times the range, so for a 600mw video transmitter, it would give you less than double the range compared to a 200mw.
But of course that depends on many other factors too, such as surrounding and how good your antenna is, and how well they are tuned to the frequency you are using.

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However, the downside of 600mw compared to 200mw and 25mw is that, it's so powerful it interferes with other channels.
If there's four people flying on 200mw and someone turns up with a 600mw, it always causes problems for everyone else's video feeds, and that person could possibly ruins the race for everyone.

You will get a better experience if you use directional antennas rather then raw power.
600mw will swamp lower powered quads (200mw and 25mw for sure) and don't really have that much benefit unless you are heading into a forest without LOS or flying away in great distances (or simply flying by yourself). Also the more power your VTX uses the more it will heat up normally.

200mw seems to be the sweet spot to fly around yourself and you get good signal up to 200-300 meters with LOS or 50-100 meters into a forest with directional antennas.
25mw is great for indoors.

I find that high humidity isn't that good for signal propagation either so you will probably have better signal in clearer days.

I agree with what Oscar and Carlos have said.

From personal experience, based on the terrain I fly I can say that for the most part the 600mW does not perform any better than the 200mW.

A lot of factors come into play, from where you are sitting or standing when you fly, what obstructions at any given time are between you and your quad, Remember, that the human body is one of the best blockouts for your video signal.

I have also ran a few 25mW Vtx's in the exact same places I fly with 600mW and 200mW and again, not a whole lot in it, The 25mW performs really well in a nice open area and no noticable difference in signal with an unobstructed view when compared to the other 2.

Getting into the trees and low shrubbery the 25 mW does break up quite a bit compared to the 200 and 600, but the 600Mw is not the be all to end all, As Carlos said, and I agree, The sweet spot is the 200mW for sure. and your signal will improve with a good matched pair of antennas.

The likes of the open clover leaf type antennas do work well, once maybe twice, if you crash and bend the lobes they still work but not half as good. They look cool, but honestly, not worth the ripoff price for what you get. I am not a lover of any particular brand of anything, but I have tried literally every brand of antenna out there and I always come back to my immeriosn RC LHCP Spironets.

anyways, that is my take on it. Buy the best VTX and antenna you can afford (I do not mean buy the most expensive, shop around, read a lot, and avoid all the sales pitch's from those selling their brand of antenna, they will tell you it will cook your breakfast if it meant a sale).

Remember, when FPV flying you 100% rely on your video signal..make you fpv setup the best you can. Get good antenna's .