How beneficial are flaps when landing with a tailwind?
Forget about tailwind/headwind for flap use.  Flaps allow landing with a lower airspeed.  And you always want a lower airspeed.  The wind itself is irrelevant except that your groundspeed will be higher or lower depending on where the wind is, and which you always want to be slow as possible.  On light aircraft you want to avoid tailwind components over 5kt. Jets are usually certified, takeoff and landing performance wise, with a max tailwind component of 10-15 kt. But in any case you are still using normal landing flap to have the minimum airspeed, giving the minimum groundspeed for a particular condition. 

The time when you may want to land with less than landing flap is when the winds are strong and very gusty (and you will be landing into wind if you aren't crazy) and you want to have a bit more energy with a bit less drag (so the aircraft conserves more energy with its inertia) during the landing flare to cope with sudden small airspeed variations from gusts, which can drop you on the runway hard before you can react if you have the flaps all the way out.