Hamilton, featuring in the race for the first time with new team Mercedes, had impressed on Saturday as he acquired pole during qualifying.
And he appeared set to have made full use of the position to dominate the opening laps, and was edging away from second-placed Vettel when his tyre failed spectacularly as he was building speed on the Wellington Straight.
The mishap dropped him to dead last, although an inspired recovery saw him pick up a handy haul of points with a fourth-placed finish.
But the 28-year-old was frustrated to have missed out on another Silverstone win, to go with his masterful wet-weather drive in 2008.
He said: "I'm disappointed by today's [Sunday] race and I would have loved to deliver the win this afternoon for all the British fans here at Silverstone.
"The race started so well for us today; I felt comfortable out in front and was managing my pace. The tyre problem was such a shame and from then it was just a case of giving it my all and battling as far up the field as possible.
"I had some good overtaking moves and, given another lap, I might have got Fernando and ended up on the podium.
"Everything considered, fourth place is pretty good after having been last on lap 10 and we'll take that."
It was also another difficult home race for Button as the McLaren driver came home in 13th after falling down the order following the second of the two safety car periods.
Button was never likely to secure his first British GP podium finish such have been McLaren's travails this season.
But the 2009 world champion was annoyed to have missed out on a top-10 finish as the decision not to make a third stop for fresh rubber backfired.
"Our pace wasn't too bad, but the last stint was particularly tough," said Button.
"After the final safety car, there wasn't a lot of tread left on my tyres, so getting them up to temperature was very tricky - we struggle with that anyway.
"Into the closing laps, I was vulnerable. All the cars on fresher tyres were fighting to get past, and as soon as I went off-line to defend, it became even more difficult to get temperature back into the tyre."