Travelling from Gatwick to London Victoria is about 30 minutes

Travelling by taxi accross London from Victoria to Paddington is about 20-30 Minutes

Travelling from Paddington to Heathrow is 15 minutes (train frequency every 15 minutes)

Gatwick to Heathrow by road should be quicker than that journey but not by much....


Time wise there is not too much difference -

but unless I was getting the taxi between the stations and unless I was happy to stay at the Hilton in Paddington.... (rather than having to walk to the station from a nearby hotel with luggage) I wouldn't see the advantage over going straight to Heathrow....

especially (as has been suggested) you could possibly go direct from Gatwick to a Heathrow hotel by pre-arranged car.....

The other question would be.... if you are not so tired - you will have little to do at Heathrow all evening of course ......!