Depends on what you want to mainly shoot with that lens: the f/4.0 is a very good landscape and short sports tele zoom; the f/2.8 shines when you shoot people, including portrait (that's why reporters love this lens), and it is of course the better choice if you want to use it with extenders. But I do not want to teach you, you surely know exactly for which purposes you want such a lens.
Btw the f/4.0 L IS USM is not only quite compact & light (like the non-IS version), feels very nice in the hands, in particular turning the zoom ring feels so light and smooth, it is a real fun to use that lens. The f/2.8 L IS USM II that I have feels extremely solid, too, but it is much more massive.
Yeah — for me it was my kid for a long time (now in university), and during that time we bought a farm and live in a rural community. (I know, weird for a tech guy.) As such, I've gravitated towards domesticated animals (e.g., ducks, horses, hounds); wild animals (e.g., eagles, deer, elk); and people or my own experience in that context (my avatar pic is from a rodeo), such as portraits around barns, warehouses, shops, etc.
For me the 70-200 f/4 original on an R6 with DLO is very satisfactory in terms of quality, especially for animals that aren't shy or things like cowboys riding a bucking bull. IS doesn't help much beyond what the R6 is providing for these contexts. That stated, I am on the Pacific Northwest, so things like humidity and rain are an issue. The lens has some internal sealing, so it's always fine (no fogging, etc.) but I don't bring it out much in the winter or spring. The IS versions are sealed better, so they remain attractive in that regard. I also powerlift, so the larger lenses are more a packing inconvenience than anything else.
But, I also have followed the guiding principle of expanding capability before replacing capability. It's tough for me to upgrade from 70-200 to 70-200 unless the advantage is overwhelming. I still have my ultra-telephoto capability to establish, for example, and since I won't justify $10k+ on a lens (I'm not making money in most cases) and I like primes (for the zen, not pragmatics) I'm waiting for that mythical 500 f/5.6 unicorn to show up. It's why my sig has all of those lenses: each provides a capability or interesting feature that I actually use in a situation that I'd otherwise miss emotionally for my day-to-day or month-to-month prosumer amusement. It's also why I'm not an RF person yet — the gains are there, but for the cash most of the time those gains (to me) are not overwhelming vs. my late-EF editions combined with DLO.
But I really appreciate everyone's input and experience, because my friends and local stores don't carry everything so I can't always test things out before buying. And renting is a real inconvenience for where I am.