Other than perhaps magazine feeding and factory loads what advantage if any does the 6 Creedmore have over the 6BR?
Going back to the original question, if you take the Lapua small primer 6.5 Creedmoor case and neck it down to 6mm to get to the 6mm version (and I assume maybe the Peterson Cartridge Co. SR primed version of same too, but I can't comment never having seen one on this side of the Atlantic) the Creedmoor is really just a lengthened case version of the BR.
Case water capacity increases from around 38gn to around 54gn or 40%. The internal ballistics rule of thumb is that if everything else is equal (bullet, barrel length, peak pressure, powder characteristics other than burning rates which will be different if optimal) then the MV % increase = the case capacity increase divided by four. In this example 40 over 4 = 10% performance improvement. So if the 6BR gives a 105gn bullet say 2,800 fps MV in a particular barrel length at peak loading, then the Creedmoor should be able to get another 280 fps making it 3,080 fps MV.
Both cartridges use identical case-head and web set-ups, both have 30-degree shoulder angles, both have adequately long necks (the 6BR longer than necessary for good performance hence the success of the Dasher whose fireforming reduces the parent case's neck-length considerably). I can't comment on case taper off the top, but doubt if they're that much different.
It adds another way-step in what is a range of competition 6mm cartridges, old and new.
At the bottom of the case-capacity ladder you have the trio of wildcats based on the 222 Rem 'family', the largest of them the 6X47mm (6mm-222 Rem Magnum) in its day the short-range BR cartridge to beat.
Then there is the 6PPC with around 33gn water capacity.
The 6BR and its improved versions at 37gn up to around 41gn water capacity for the Dasher
6mm International at around 44gn capacity (the 250 Savage case necked down with other changes) and a steeper shoulder angle than the BR and Creedmoor at nearly 35-deg in the Harvey Donaldson version. This was touted as a possible alternative to the BR in some disciplines such as CISM 3-position shooting and nearly adopted by the US Army team many moons ago.
6mm Swiss Match and 6-6.5X47mm Lapua wildcat - very similar designs but the older 6SM uses large primer brass and is therefore inferior to the Lapua case based version. The 6SM was
highly developed in Europe for 300 metre shooting, but probably never attained the widespread use or popularity of the BR which has been king for many years in Europe in the important 3-position ISSF 3-position 300 metre discipline. Around 46gn water capacity for the Swiss Match and 1.5-2gn more for the Lapua based case version. The latter has been 'improved' through fireforming on occasions too as per the 6BR to Dasher for another couple of grains capacity and is usually called the 'Long Dasher'. There are proponents of the 6-6.5X47L in mid and long-range BR and it was thought for a while it would seriously challenge the Dasher. Although taking more than a few records, it hasn't dislodged the Dasher though and seems to be losing ground these days.
That brings us to the Tubb 6XC, 6mm Creedmoor, Whitley 6mm SLR (a reformed 243 Win), 243 Win, and 6mm Rem in ascending size order with around 50gn water capacity up to 54gn. The XC, 243, SLR, and 6mm Rem use large primers as does the Creedmoor in factory Hornady form (but most people using it for precision shooting will reform 6.5CM small primer brass as mentioned). The XC, SLR, and Creedmoor have long, or long enough, necks allied to 30-deg shoulder angles and are efficient modern designs. SR primer case-head brass will be stronger and take higher pressures for longer than the others though. All feed well from magazines. the 6mm Rem has an improved wildcat version called the 6mm Crusader - a very capable mid/long-range F-Class cartridge (while the barrel lasts which isn't that long in full-pressure loadings). There are many who look down on the 243 with its relatively short neck and 20-degree shoulders, but John Whidden keeps winning long-range sling shooting matches and championships with it. (I do prefer the Robert Whitley's 'SLR' variation though where a pass through the sizer die bumps the shoulder back to produce the XC's longer neck and 30-degree shoulders.)
Then finally, although I'm sure I'll have missed something, there is the 6-284, a necked down 284 Win or 6.5-284 Norma case and a one-time favourite of long-range precision shooters. Superb ballistics but a barrel burner. A very strong and efficient rebated-rim case design with 35-deg shoulders.
All of these cartridges have been or are still being used in competition in original factory or improved wildcat forms. With fast-twist barrels and heavy bullets everything from 6mmBR up is 1,000 yard capable.
Magazine feed aside, they all have pros and cons. As a general rule, the smaller the case and powder charge to bore size, the easier it is to shoot, the less stress put on the action and barrel and therefore the opportunity to attain very high precision increases. The larger cased versions give higher MVs and therefore with any given bullet weight less wind drift, also often allow the cartridge to work better with heavier bullets (115s for 6mm instead of 105-108s). This can be particularly important to prone shooters with their longer courses of fire and targets pulled for every shot allowing far more time for the wind to change compared to a fast 5-round BR string. As more powder is burned, efficiency drops as does barrel life, heat and recoil increase, and while it may not be impossible to get as good precision as in the smaller cartridges, it definitely becomes harder. So as always it is swings and roundabouts - gain on one, lose on the other; put crudely precision v external ballistics.
So putting PRS and other 'tactical' type shooting to one side with their need for snap shooting, low recoil, flat trajectories, high-cap magazines giving flawless operation etc, the 6mm Creedmoor will generally give higher scores than the 6BR / Dasher in mid-range F-Class and other such disciplines if the wind is blowing all over the place, but not necessarily so in calmer conditions. At long ranges, it will outperform the BR (but in its turn be at a disadvantage to 6.5s and sevens) in such matches in all but the gentlest of conditions.
In BR, as a rule, things are reversed unless the conditions are horrible as the BR's inherent outstanding precision beats almost anything if the shooter can avoid suffering a wind change mid string.