It should still be the same Mesa driver, but if that card is brand new, then you may need to install a later versions of the Linux Kernel and Mesa. There are a couple of PPAs which supply newer versions of these items, but I'm not sure which PPA is best. Keep in mind installing any software from PPA's is a bit risky as you are relying on other community members choosing proper & stable compile settings when creating the newer versions of the Linux kernel and Mesa.
Sometimes it can take a few years before the open source Linux AMD GPU drivers get really good for a new GPU.
The drivers from the AMD website should not normally be used by most people because the open source drivers are usually much better performing. The core Linux Kernel driver portion is shared by both the free open source driver and the AMD proprietary driver. It is possible the driver from the AMD website may be a slightly better choice for a new GPU architecture until AMD updates the free open source Mesa drivers. The proprietary AMD driver is usually only needed if you need special OpenGL features for professional graphics apps or for the GPU compute feature.
So is this two different GPUs you are talking about? The W7500X and the HD7950?