In summary the Tamron SP 70-300MM F/4-5.6 Di VC USD is a very good lens, especially considering the cost. In the center of the frame image quality is very similar to that of the Canon EF 70-300/4-5.6 IS USM at all focal lengths and apertures, however at 300mm the Tamron shows better image quality in the corners of the frame, especially when the lenses are used wide open with a full frame DSLR.
At 135mm and 70mm there is little to chose between the Tamron and Canon lenses, Both are sharp in the center and still good out to the corners of the APS-C frame. At 135mm the Tamron lens shows slightly better image quality in the corners of the full frame image.
Both lenses show similar control of chromatic aberration, which although present is at a low level. Both show some vignetting, especially wide open on a full frame camera, but no more than you would expect (around a stop) from a lens of this type. Both lenses have equally effective image stabilization of around 3 stops.
In general use on an APS crop sensor body most users would not see much difference in image quality in smaller prints (up to, say, 8x10). However the Tamron is a little better in the image corners at 300mm which would show up on full frame images and in larger prints, so overall (though mainly based on edge performance at 300mm) I have to award the Tamron the "better image quality" prize.
If both lenses were the same price and came with the same features and accessories, the contest between the Tamron SP 70-300MM F/4-5.6 Di VC USD and Canon EF 70-300/4-5.6 IS USM would be close. However the Tamron is currently cheaper, it comes with a lens hood, it has full time manual focus and a distance scale and it comes with a 6-year warranty. So at this point I think I'll have to agree with Tamron's claim that the is the best lens in its class and very good value.
I can definitely recommended the Tamron SP 70-300MM F/4-5.6 Di VC USD if you're looking for a telephoto zoom for a Canon, Nikon or Sony DSLR. Not an option though for Pentax, Sigma or Olympus users.