It depends on what you are shooting and how you are stabilizing the camera. For theatrical shooting I use a monopod and an EF 70-200mm f/2.8 L IS II. I still have to time my shots when the performers are relatively still.
The image stabilization on the EF 70-200mm f/4L IS is good for 3-4 stops. On a crop body without IS, at 200mm you wouldn't want to shoot handheld below about 1/320 sec. With IS, you could drop down to about 1/40 or even 1/20 sec. But that requires that your subject isn't moving.
If your subject is moving very fast, the shutter speed needed to freeze the motion is going to make IS superfluous. Shooting sports like football or soccer from the sideline requires a shutter speed of around 1/500 sec. or faster. The horses and dogs you shoot can move faster than most athletes, and you are probably trying to shoot them from closer which necessitates an even higher shutter speed.
If you are shooting from a stable tripod, the extra stop of the f/2.8 version would probably be more valuable to you, if the lens is sharp enough for your needs at f/2.8. You will also be able to get more bokeh with the f/2.8 than the f/4 lens.
You will not gain anything in terms of shutter speed or IS in moving from the EF 24-105mm f/4 L IS to the EF 70-200mm f/4 L IS. The 70-200 is a little sharper and has more reach, but gives up even more on the wide end.
There are a few other things to consider:
Some people think the four versions of the EF 70-200mm "L" are all pretty much the same lens with various apertures and with or without IS "added". They are not. Each lens is an entirely independent design and reflects the state of materials and lens design technology available at the time they were released. The newer F/4L IS is sharper at f/4 than the older f/2.8L. Even the f/2.8L IS design, over a decade old having been introduced in 2001, is not as sharp as the f/4L IS.
Of course the king of the hill of Canon 70-200mm "L" glass is the EF 70-200mm f/2.8L IS II. Introduced in 2010, it is at an entirely different level in terms of sharpness over the entire focal length and aperture range. It is also considerably more expensive. Back in 2010 I needed a good, constant aperture telephoto zoom lens. I considered all of the options you are now considering, plus third party lenses available at that time as well. I came to the realization that if I bought anything less than the f/2.8L IS II I would always wonder if I should have bought it instead. I decided to wait a little longer and save enough to get the f/2.8L IS II. That is one of the best decisions I've ever made in terms of photo gear. It is the best zoom lens I have ever used. I had to save for quite a while to be able to buy it. Many meals that could have been eaten in restaurants were cooked at home. Many other things I wanted were put on the back burner. The cost of this lens was totally forgotten when I looked at the first images I shot with it. It is worth every penny I paid for it. I consider it some of the best money I have ever spent on anything.