That HP driver could be worse in term of performance than generic one from AMD?
I wouldn't imagine so, or at least the difference probably wouldn't be significant, especially since we are talking about a low baseline (i.e., the absolute value of every % increase would be very small)
Like all vendor binaries, it's been QA'ed and probably tweaked a bit for the specific hardware components found in the model it was made for. Performance is probably similar to generic ones from AMD's download page, perhaps a bit less in cases where AMD's more recent drivers includes optimizations in certain games, but I wouldn't really recommend using a non-vendor-specific driver since it might affect other things other than GPU performance (such as behavior/compatibility with sleep/wakeup, maybe customized power-saving mechanisms that comes with the vendor-specific driver, etc).
The chipset driver and graphics driver (w/control panel) come as separate drivers, with the chipset drivers being much smaller (40mb vs 200mb). The chipset drivers are much newer (May 2013) while the graphics driver is surprisingly dated (Jul 2012).
I don't know, maybe I will give it a try with AMD's latest catalyst. If I get time to re-bench a few titles with just a single stick of RAM, then I'll also throw in a test using AMD's latest driver.