I own a 70-200 f4 IS, and last year I rented the 70-300L to try it out.
Let's start by saying that I absolutely love my 70-200. It's small, light, gives outstanding IQ, has lightning fast focus and does everything well. I bought it several years ago to replace a f2.8 non-IS version that was too heavy and bulky for me.
The only reason I wanted to try the 70-300L was for the extra reach. I sometimes photograph dog sports, and the 70-200 is often a bit short. And I was prepared not to be impressed by the 70-300L because I love my 70-200 so much.
More fool me. I was blown away by it. It's a hugely impressive lens in every respect: IQ, focusing speed, and the IS is amazing. Not to mention the extra reach, which I loved.
Bottom line: if I could bring myself to sell the 70-200, I'd buy the 70-300 in a heartbeat. As it stands, I'm saving up for it.
If I were starting from scratch today and had to choose one of the lenses, I'd choose the 70-300L.
PS I have a 1.4X that I use with my 70-200, but the combination is not as good as the 70-300L.