How well does the Canon 5D III / 6D HDR mode rate against other HDR modes? i.e. using a computer + HDR software?
This is ultimatly a fairly subjective question. I think it does a pretty good job on my 5D Mark iii most of the time. It's also worth noting that you can have the 5DM3 save the individual files used (including in RAW format) so that you can always use software later even if you decide you don't like the JPEG that it produces. There are also multiple styles and multiple levels of EV range that you can use for capturing the images.
If you click here, you can see examples of the HDR mode in the 5D mark III. There are actually several presets, so it gives you the freedom to choose one you like. Of course you have less freedom compared with software like photomatix.
It is not possible to say which one is better, since this is also a matter of taste. I think its a trade off between ease of use for the canon vs adjustability for the software solutions. But both could give satisfying results.
And of course these cameras also have easier settings for bracketing which is also an advantage in itself.
You will have far greater control of your final image if you take several pictures and combine them later using an HDR post-processing tool. You'll have the choice of making it photo-realistic (like the built-in camera modes) as well as going all "cartoony" like many on the internet like. But you also have the ability to get the shot looking exactly like you want.
You also have the option of using as many shots as you want, covering a greater dynamic range. I don't now about the 5DIII and the 6D, but many Canon camera's only take 3 images, and some of them, like the 5D and 40D limit you to +-2 stops. (Magic Lantern can be used to get past this limitation.)
Of course, all this takes time, but if your goal is the perfect shot, you'll want to use post processing tools. If your goal is to just get the scene now, capturing more of the dynamic range present in the scene than the camera can handle, use the built-in mode.