Vettel, on pole three times in four races this season, dominated a sunny afternoon by lapping the Circuit de Catalunya with a best time in one minute 19.965 seconds ahead of Australian Webber.
"I think we've proved today that our changes work," said Vettel, whose team have brought various improvements to the first race of the European season.
"It's good to be fast but I think the McLarens are quick here. I don't think they were pushing hard this afternoon and I think it will be much closer tomorrow."
Schumacher, the sport's most successful driver who has struggled to match his young German team mate Nico Rosberg so far in his comeback season at the age of 41 and after three years out, was third in both sessions.
The second was stopped briefly when Germany's Nico Hulkenberg crashed his Williams.
Reverting to the chassis that he had used in pre-season testing, and armed with a longer wheelbase and new aerodynamic parts, Schumacher found the changes very much to his liking at a circuit that has handed him six of his 91 wins.
"Today I felt much more comfortable in the car and feel that it is behaving more consistently," said the seven times champion. "In my view our upgrade is a step forward but we have to wait and see how big the step is."
Schumacher was 0.582 behind Hamilton's first session pace and 0.792 slower than Vettel's in the second.
Rosberg, who has been on the podium in his last two races while Schumacher retired in Malaysia and laboured home 10th in China, was seventh.
"I'd say today things are back to normal as an extension of what we saw after Malaysia with Shanghai being a blip," said Mercedes team principal Ross Brawn.
"It is very early in the weekend but we saw no areas of concern today which we did have in Shanghai."
Fernando Alonso, the double world champion preparing for his first home grand prix as a Ferrari driver, ended the afternoon in fourth place after setting only the eighth best time before lunch.
He said he was reasonably happy with new developments brought to the car, including a rear wing that benefits from a similar "F duct" system used by McLaren.
"We will decide this evening whether or not to also use it in qualifying and the race," said the Spaniard. "The system worked well but obviously...we still lack a bit of experience before we can get the best out of it."
While Friday practice provides a morale boost, Saturday's qualifying will be crucial with the winner in Barcelona starting on pole position for nine years in a row.
Austrian Christian Klien made a return to Formula One after signing a reserve driver role with newcomers Hispania (HRT), taking Indian Karun Chandhok's place in the morning.
Britain's Paul di Resta also returned to the track for Force India, the tester replacing regular racer Adrian Sutil for the morning only and lapping quicker than the team's other driver Vitantonio Liuzzi.