When it comes to bare lens IQ, the 70-200 II is better. There is something with the natural vignetting at f/2.8 and colors that help the subject pop. AF speed is good. I spend less time post-processing 70-200 II files than the 70-300L's. The difference in look between f/2.8 and f/4-f/5.6 is noticeable especially for portraiture and sports.
If you want to shoot above 200mm a lot, then the 70-300L comes into its own. The 70-200L works well with the 1.4x, but then it loses a lot of its advantages over the 70-300L. For some reason, I don't like the images taken with the 70-300L on the LCD on the back of the camera but I end up liking them when I open them in LightRoom (mostly due to color reasons, so much so that I force myself to chimp less when using the 70-300L). When comparing the 70-200 II with the extender to the 70-300L, the IQ (sharpness, colors) is similar, but the 70-300L is lighter, more compact, better at 280/300mm and having a wider range (70-300 vs. 98-280) while costing up to a stop in speed. The 70-300L AF is a little bit snappier than the 70-200 II + 1.4x.
What it comes down to are well known trades between weight, size, how often you intend to use the 200-300mm range, and whether you need to use this for low light applications, where every stop counts. The 70-300L is one of Canon's best travel lenses, and I would also leave the 70-200 II home when traveling (after lugging the 70-200 II to Disney for a week while trying to watch two young kids). That we would opt for the 70-300L over the 70-200 II for travel is a testament to how good the 70-300L is because we don't lose much in IQ but we gain a lot in versatility. However, a lot of us will also bring a fast prime to complement the 70-300L for thin DOF or low light use for portraiture or for indoor applications. With the 70-200 II, you might be able to skip the fast prime and make do with the f/2.8.