With today's powders & bullets, the 300 H&H is better than ever, but I guess that is true for the '06 as well.
Both are competent & real performance is virtually indistinguishable, but whatever edge there might be in performance surely lies with the H&H, albeit, not a big edge.
And the "cool factor" surely lies with the H&H too. 
MM
Guessing that you figure "cool factor" differently than I do.
The 30-06 has oodles more history, and is associated with more of the coolest people than any brit round can lay claim to. Only the tweed coat/bwana crowd would ever attempt to debate the point.
I currently have an H&H in captivity, and hope to hunt with it this fall. It was the rifle package that drew my attention though, and not the chambering.
We often forget that it was the British and Europeans who developed most of the famous calibers specifically for hunting in the days when untold hundreds of thousands of animals were taken in the colonies.
Many, like the 300H&H and 375H&H are still recognised as the best at what they were designed for. Clean kills with little meat damage.
Other 300 mags were poor attempts to copy the H&H and marketing to the US hunter that they must be better because they made them "faster".
Unfortunately the African animals did not have the benefit of the American education system and never learned to read ballistics charts and don't understand that more "speed" and "energy" are needed to kill.
American game animals on the other hand must understand it or there wouldn't be so many hunters there with Ultra mags, Weatherbys, etc.These hunters are quite easy to spot at SCI conventions as they usually have a scar over one eyebrow and a nervous "wink" - and are always "ballistics experts".
Those old greats have improved with modern powders, bullet construction, and optics - but they have yet to be beaten.
America can take credit for the 30-06 and maybe should have stopped there.
Somehow they got confused between "hunting" and "varmit shooting" at 300+yds with high powered scopes high speed calibers and bullets meant to destroy the intended target as it wasn't edible anyway.
The mauser 7mmX57 was also a great African classic with books written about it. H&H tried to get into that huge market by "improving on it".
The 275H&H was introduced by Holland&Holland in 1912 using the 7mm bullet with a shorter version of the belted case of the 375H&H mag introduced the same year as the .375 Belted Rimless Nitro-Express.
The .375 H&H was intended for dangerous African game animals, while the .275 H&H was intended for longer range shooting of African antelope and Red Stag in the highlands of Scotland.
Western Cartiidge Company offered USA loadings of the .275 H&H Magnum in 1925 with the .300 H&H and the .375 H&H. The .275 H&H was omitted when Winchester started chambering their Model 70 rifle for the other two in 1937.
The .275 H&H offered little ballistic advantage over the 270. U.S. ammunition production ceased during 1939 and it died - only to be ressurected as the wildcat 7X61 Sharp & Hart in 1953 and finally the 7mm Remington Magnum in 1962.
Over the years We have had safari clients with every conceivable magnum you can imagine including new calibers sent here with outdoor writers to "test" them in Africa. most were disappointing and some were disasters.
I'm not sure why, I started with hundred year old calibers but I do know the reason I still have them is because they have never failed me, I am confident shooting them, and I have been extremely disappointed with with the newer, faster, calibers. I have had the opportunity test both old and new. I love the new rifles being built today - but I'll stick with my "old" proven calibers.
One day hunters might stop listening to the hype and and choose calibers based on what really works - but I may not live long enough to see it.
