It's the very same driver. And one big selling point of NVIDIA has always been that you could run CUDA on your consumer card too - unlike AMD where their compute stack of the day would only run on a select few cards of the last generation.
It is mostly the same but the support that the DCs are getting are completely different. Nvidia can and does give them slightly tweaked drivers. Data centers ship a set of enterprise distros with tightly managed set of software that's been vetted. Since there are no binary guarantees in Linux, anything can happen with consumer hardware.
Source: once worked as an student system admin in my university HPC center.
