The ones with the better 'eco' & fuel ratings tend to be those with the least rolling resistance - less resistance = less wear = less fuel used.
So if you want a tyre that lasts a long time (although what do you consider high mileage?), is a bit noisy and gives you 1 or 2mpg better, and you don't need the last word in grip (i.e. for track days or hooning) then go for it.
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Bit of further info around the new tests though, as each manufacturer grades their own tyres within their own range. There are however, ranges that they're supposed to stick to, so a manufacturer might change a compound/pattern slightly if their tyre is on the cusp of a better rating, but only if it doesn't affect performance in another category.
...and there are plans to change the label this/next year
Wet Braking/Grip (not wet weather, so it may stop well or be resistant to aquaplaning at 40mph, but not turn well in the wet)
3m per level increase when braking from 50mph to 12mph
can't find any info on what scores are provided by the 2nd part of the test (wet grip)
Tyre with an A-grade stop in 40m, vs a tyre with F-grade stopping in 55m - quite a big difference (it's measured up to G, but it's illegal to sell a G-rated one)
Fuel Efficiency
1.25% fuel use increase per level
Assume a tyre is supposed to last 40,000 miles, the difference between a A grade and a G grade is about 80 litres over the full life of the tyre may be 1 litre saving every 500 miles (about £1.30 per fill), but the difference between an A grade and B grade may only be 1 litre saving every 2500 miles (£0.25) per fill up).
Also, don't forget that this is only relevant with a specific manufacture, so an B-rated tyre from Pirelli may give better fuel economy over an A-rated one from Michelin. But that information is never shared as it's commercially-sensitive.
External Tyre Noise
3dB per level increase above the 'future European mandatory limit'
dB shown is measured from 7.5m away at 50mph, and the black bars show how many steps above the future noise limit they are - sometimes lower rolling resistance tyres are quieter.