As you've probably discovered the Wellcome Trust Genome Campus in very much Cambridgeshire not Cambridge itself. In fact, it's about a third of the way to Stansted!
Cambridge airport is small, so your time from plane to kerb will be very quick. Stansted, not so much...
By taxi, it's about 12 miles from Cambridge airport to Hinxton. If you've pre-booked a minicab, it should take about 25 minutes. A metered local taxi would be about £30, but you can probably negotiate a lower fare if you pre-book a minicab. I'm not sure there are taxis waiting at the airport normally, as it's small, so you might need to book anyway.
By bus from Cambridge airport, you'd need to get a bus into town (11 or 77), then either a Citi 7 to nearby, or the Park and Ride bus to the Babraham Road Park-and-Ride then the 7A to Hinxton. Plan for about 1.5-2 hours
From Stansted, in a taxi it'll only be about 30 minutes, as it's a motorway almost the whole way. Downside is it's 20 miles, with an airport pickup surcharge, so it'll be quite a bit more money than Cambridge. My hunch is about £45 for a local taxi, less if you pre-book a minicab and negotiate something.
Public transport from Stansted to Hinxton isn't great though. You can ask Google Maps, but the fact that it often suggests taking a train from Stansted to London then train London to Cambridge then bus then walk should give you some idea... There doesn't look to be any local buses covering the route, so you're pretty much stuck with train or long distance coach to Cambridge (passing Hinxton on the motorway without stopping), then basically the same local buses as you'd have got from Cambridge airport.
If it were me, and the flight costs were about the same, I'd suggest flying to Cambridge, bus into town, enjoy a little bit of time in the city centre, then taxi out to Hinxton. Likely to be much more enjoyable than via Stansted, and given the queues you often encounter at Stansted it may even be quicker even with an hour's sightseeing / eating / drinking in Cambridge!