Smith was the only possible choice for best afield at Lord's, his double-maiden century empowering the tourists to square the series at 1-1.
At Edgbaston, Smith was dismissed in single figures in both innings.
It was the first time the 26-year-old suffered the fate since scoring the first of his 10 Test centuries.
Root hit the winning runs for England, having top-scored with 63 in their first innings.
As a result the Yorkshire batsman climbed to third sport on the International Cricket Council's Test batting rankings, rising to 882 points.
World No.1 Smith is on 912 points, meaning there could well be a changing of the guard depending on how both men fare at Nottingham and The Oval.
"I'm not seeing it as a competition in any way like that," Smith said after his masterful 215 at Lord's.
Root has offered similarly diplomatic responses when asked about the burgeoning rivalry.
"I don't think I'll be facing him much, so I'm more interested in the battle with the seamers they've brought across and Nathan Lyon," he said.
"Steve's had a fantastic couple of years where he's done very well for Australia and we want to see the back of him pretty early."
There are compelling arguments that both fresh-faced stars deserve the mantle of top dog.
They score freely, stroking boundaries and picking the gaps with aplomb.
They are clearly capable of changing the course of a Test.
And they are both incredibly selfless, part of the reason they are set to lead Australia and England in the 2017-18 Ashes.
Meanwhile, Josh Hazlewood and Nathan Lyon climbed the ICC bowling rankings after the third Ashes Test.
Hazlewood, leading-wicket taker of the Ashes, is up three places to 14th while Lyon is 15th.
Mitchell Starc's knock of 58 in the second dig helped him draw level with teammate Shane Watson in ninth spot on the Test allrounder charts.
-AAP
