One has a longer reach than the other. One has a fixed aperture throughout its focal range, the other becomes smaller as you go in closer. Based on this information, you can decide on which lens you need.
I used to own the 70-200 f/4 twice before and both times sold it. My 2nd piece i sold it for the 70-300L due to the tele end at 200mm is a tad short on FF. Aperture wise doesn't matter to me as the 70-300 has all the features i need.
Get the 70-200 f/4 IS if
1. you absolutely hate the zoom-out design of 70-300.
2. requires constant f/4 aperture
3. needs something lighter
4. contended with 200mm max
Understand that the discussion is on the 70-300 L, how abt the non L version of it? the pricing is a point tat attract me too. Din have th eopportunity to test shoot using the Non L version. Is the IQ commendable too?
This picture is taken with the Sigma 150-500mm, at 500mm wide open. It is a little soft at max focal length as compared to the Canon, but I find it acceptable already, especially given the pricing.
That said, I think if budget permits, the Canon 100-400L makes quite a good choice for birding. It is much more compact as compared to the Sigma, and can be carried around easier. I certainly won't want to carry the Sigma for street photography. Of course the 70-300mm wins both of them hands down, but for me, a FF user, the range is slightly short
Get the 70-200 f/4 IS if
1. you absolutely hate the zoom-out design of 70-300.
2. requires constant f/4 aperture
3. needs something lighter
4. contended with 200mm max