Hamilton was quickest in a cloudy morning session with some rain, clocking a lap of one minute 21.537 seconds while Bottas was second, but the tables were turned in the dry afternoon running.
The Finn pulled out a 1:21.406 while Hamilton - who will oust Vettel from the championship lead he has held all season if he wins the last European race of the year on Sunday - was 0.056 slower.
Vettel, seven points clear of his British rival, was third in both sessions with the gap closing in the afternoon to just 0.140. He had been more than a second slower in the opening 90 minutes at Ferrari's home track.
"It's been a good day, a clean day," said Hamilton, who won in Belgium last weekend and is favourite to take a record 69th career pole position on Saturday.
"We got the running done, we got through our programme with no problems. The car seems nicely balanced here. We just have some work to do to eke out a little bit more performance.
"It looks quite close between us and the Ferraris, so I anticipate it's going to be similar to the last race in that sense."
Ferrari Kimi Raikkonen was fourth fastest in practice one and two.
The Ferrari fans will take heart that Mercedes also looked strong in Belgian Grand Prix practice at Spa last weekend. Ferrari closed the gap, with Vettel on Hamilton's tail throughout the race.
They will also be buoyed by the fact that no driver has won two races in a row this season.
"I think going into this weekend we need to be fair and say that the track layout suits Mercedes, but the performance that we showed last week gives us hope," Vettel had told reporters on Thursday.
The Red Bull pairing of Australian Daniel Ricciardo and Dutch teenager Max Verstappen followed in fifth and sixth places, but both face grid penalties after Saturday's qualifying because of power unit changes.
Verstappen will have a drop of 15 positions while Ricciardo, who finished third in Belgium last weekend, has a 20 place penalty.
In a surprising move up for struggling McLaren, whose Honda engine has been short on both performance and reliability, Belgian Stoffel Vandoorne was seventh quickest and Fernando Alonso eight. Alonso also has a 35-place grid penalty, however.
Spaniard Carlos Sainz, who also has a 10 place grid penalty, ended the second session early when his Toro Rosso suffered an engine failure.
The virtual safety car was deployed near the close when Kevin Magnussen's Haas stopped on track with a broken suspension.
(Reporting by Alan Baldwin; Editing by Larry King/Keith Weir)
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