What NVIDIA doesn't support as of yet is Asynchronous Compute, which is as part of AMD's GCN architecture and is proprietary. Some developers are developing games to use that feature since AMD is being more open about it. AMD's GPUs have the advantage of implementing it at hardware level, while all NVIDIA can do for now is somehow make asyc compute instructions work on their GPU through software patch, and in future may provide a hardware implementation of their own that would be capable of doing the work of async compute engines. That is why AMD pulls up ahead by a good margin in Ashes of Singularity DX12 benchmarks.
To be clear, neither AMD nor NVIDIA support all features of DirectX12 as of yet. That is set to change with their upcoming GPU series, which is not quite far off. According to reports, both the companies are going to launch then new GPUs this summer. So I guess you should wait a bit since the new GPU will be on 14/16 nm architecture, which will allow greater power efficiency along with the goodies new GPU lineup tends to have. Polaris seems to be promising and if it's claims holds merit, AMD could be back in the desktop GPU segment with a bang! Also since most of the next-gen games will eventually use directX12, newer GPUs will perform better and offer more support for directX12 features.
As for your other concern, the price should be nearly same, although launch prices tends to be a bit high.
Rough estimate of release: NVIDIA: may; AMD: June (As I gathered from articles at wccftech)