The only way you are going to know for sure is to do it by yourself. The answer is not definitive and is dependent on your system... how many cores, what else the OS is doing at the time etc.. Yes, ArcGis and QGIS are that close that it sometimes makes little difference which one you use.
The bigger question is why buy ArcGis when QGIS is free?, the answer is the variety of tools, the API and corporate support also ArcGis is a recognized industry standard, having that on your resume is a define bonus, but then again QGIS is becoming a standard too.
Esri has been working on the low-level code for many years and many very intelligent people have optimized the nuts-and-bolts so individual processes are likely to be faster, but with that in mind the lower level code hasn't been changed much in years and it's possible that it's getting a bit antiquated.
I have bench marked QGIS against ArcGis for a capture/revision project and found that it was no slower than ArcGis and that was at version Wroclaw against ArcGis 9.3.1. Recent advances in multi-threading and 64bit (if you have a 64bit OS) coupled with the fact that QGIS will run on Linux (generally faster than Windows). ArcGis is still a single threaded application, on a CPU with less cores and higher speed ArcGis generally will perform better than QGIS and conversely with more cores and lower speed.
I am a dedicated, daily user of ArcGis but find that QGIS has its uses (frequently). I say use both and get familiar with both. They are both excellent packages and are both worth using. If you must concentrate on learning one of these packages over the other then there is a big decision to make with pros and cons for each - well worth another SE question! Personally I'd like to see an intelligent discourse on "I have the opportunity to learn ArcGis or QGIS, which one should I concentrate on?"