The tentative 16.04 release notes mention, "The fglrx driver is now deprecated in 16.04, and we recommend its open source alternatives (radeon and amdgpu). AMD put a lot of work into the drivers, and we backported kernel code from Linux 4.5 to provide a better experience. When upgrading to Ubuntu 16.04 from a previous release, both the fglrx driver and the xorg.conf will be removed, so that the system is set to use either the amdgpu driver or the radeon driver (depending on the available hardware)."
Canonical will not be supporting the fglrx/Catalyst Linux driver in their archive beginning with Ubuntu 16.04 Xenial Xerus. There's nothing stopping anyone from downloading the latest release from AMD.com and installing the driver directly or generating the Debian packages from that -- assuming upstream Linux kernel and xorg-server compatibility -- but it's not being supported by upstream Ubuntu.
For most users, the open-source Radeon graphics stack is "good enough" particularly if you just care about desktop use-cases, video playback, and light gaming. However, currently the open-source Radeon Linux graphics stack only supports OpenGL 4.1 rather than OpenGL 4.5, the open-source OpenCL compute stack leaves a lot to be desired, and there are other current open-source AMD driver limitations that for some users will make switching to the open-source driver a regression in experience and performance.
The NVIDIA Linux driver will remain in place for Ubuntu 16.04. The news of Ubuntu deprecating fglrx/Catalyst isn't entirely a big surprise since around the middle of the year is when AMD is expected to roll out their new Catalyst Linux driver based on the AMDGPU kernel driver. There, however, that support is still expected to be limited to GCN 1.2+ GPUs at least initially so it won't be of any relevance to the vast majority of AMD Linux customers at the moment unless GCN 1.0~1.1 gets tidied up soon for AMDGPU.