Getting both of these lenses is of course ideal (and very highly recommended), but what if your budget allows for only one? Which one should you get? On the surface, the Canon EF 70-200mm f/2.8L IS II USM Lens and the Canon EF 100-400mm f/4.5-5.6L IS II USM Lens are quite different in their specs, but with the addition of the Canon EF 2x III Extender creating a 140-400mm f/5.6 IS lens from the 70-200 whenever desired, these two options quickly become rather close in primary specs.
In making this decision, the first question to ask is: "How important is a wide aperture to you?" If you are shooting action in low light, especially indoors, the f/2.8 aperture is going to be very important. If you need the maximum background blur in the 70-200mm focal length range, the f/2.8 lens is the better choice. If you simply need the 70-100mm focal length range, the 70-200 is the right choice as the 100-400 can't do that.
While the focal length range should play strongly into the decision making process, the 2x extender evens out the playing field between these two lenses. One of the first concerns I have when adding an extender is the impact to image sharpness.
Overall, these lenses are both so sharp that image sharpness is not a significant differentiator over the native focal length range overlap. The 70-200 of course needs help from an extender to cover the 200-400mm comparable range. Adding a 1.4x extender to a great zoom lens will cause relatively minor impact to image sharpness, but a 2x extender generally produces a noticeable contrast and resolution impact and that is the case here. The 70-200 performs quite well with the 2x, but the difference in across-the-frame sharpness is noticeable with the 100-400 showing a moderately strong advantage in the 400mm comparison. AF performance/speed also takes a bit of a hit with this extender in place.
These two lenses are quite similar in size, weight and price ... until the 2x extender is added to the equation. The 2x adds 2" (50mm) to the length, 12 oz (340g) to the weight and roughly 20% to the cost.
Thus, if the longer focal lengths are going to see significant use, the 100-400 L II has the overall advantage including smaller size, lighter weight, lower price, better AF performance and better image quality. Applications I commonly use the 100-400 L II for include wildlife, landscapes and big-field daytime outdoor sports photography.
When the wider aperture is needed and/or the 70-200mm focal length range is preferred, the 70-200mm f/2.8L IS II is my preference. This lens is an excellent choice for portraits, indoor events (including weddings), indoor sports, and many more similar uses. When longer-than-200mm focal lengths are needed only infrequently, adding the 2x to the 70-200mm lens can get the job done.
Again, the ideal Canon kit will have both of these lenses in it, but for those that must choose between the two, there is usually a best choice.