Mantle is useful only if your games support it, and right now it's anyone's guess whether Mantle will actually catch on, especially with the upcoming DirectX 12 rumored to enable the same benefits that Mantle brings. Get the 280X if you are into mining, and get the 770 if you are strictly gaming.
However, I would not recommend buying a GPU at the moment. The 280X(7970) is old stuff and is bound to be surpassed soon enough, and the 770(680) is almost two years old in actuality. Not saying that the 280X and the 770 aren't great GPUs, but I'm saying that you could probably get something much better for your money in the next couple of months.
If you haven't heard of the upcoming Nvidia 800 series cards, google and read up on it. The new architecture could bring benefits all around, and so far is looking impressive even on 28nm nodes. The high end 20nm GTX 880/870's release date is still anyone's guess, some expected them to be out Q1 2014, and some are now saying 2015. My speculation is that it will hit the shelves before Computex 2014, or by the end of summer at the LATEST. If you don't need a graphics card RIGHT NOW, I would sit back and wait a few months. It's not like there are amazing deals at the moment, they are simply selling for slightly above MSRP.