The basic question is one of focal lengths as compared to angles of view (AoV) when using the same focal lengths on cameras with different size sensors. The AoV will also affect needed shooting techniques, particularly at the very narrow AoVs given by very long focal lengths. The different maximum apertures will also have a bearing on the decision.
To understand the angles of view, we can use the EOS 650D's 'crop factor' to express the angle of view a particular lens would provide if a lens with 1.6X the focal length were mounted on a full frame camera such as your EOS 6D.
If you use the EF 24-105mm f/4 IS on the 6D and the Sigma 150-600mm f/5-6.3 OS on your 650D you'll get an angle of view with the 650D + 150-600mm that would look like a 240-960mm lens used on the FF 6D.
On the other hand, if you use the EF 24-105mm f/4 IS on the crop body, you'll get and angle of view that looks like a 38-168mm used on a FF camera.
In terms of angle of view, your choice is between:
- A 24-105mm AoV with the 6D + 24-105mm and a (FF equivalent) 240-960mm AoV with the 650D + 150-600mm
- A (FF equivalent) 38-168mm AoV with the 650D + 24-105mm and a 150-600mm AoV with the 6D + 150-600mm
The first option provides the widest range of coverage. 24mm on the 6D gives a diagonal AoV of 84°. 600mm on the 650D gives a diagonal AoV of only about 2.5°. There is a large "gap" in between 105mm on the 6D (≈ 23° diagonal AoV) and 150mm (240mm FF equivalent) on the 650D (≈ 10° AoV). That 105mm to 240mm (FF equivalent) AoV is an area many find extremely desireable.
The second option provides less total range of angles of view, but there is no gap between each camera/lens. In fact, there is a bit of overlap. 24mm on the 650D gives a diagonal AoV of ≈60°. 105mm on the 650D is ≈14°. 150mm on the FF 6D gives an AoV of ≈16°. At 600mm on the 6D an ≈4° AoV is provided.
The next thing to consider is maximum aperture. The EF 24-105mm has a constant maximum aperture of f/4 throughout its range. The Sigma 150-600mm has a maximum aperture of f/5 at 150mm and that narrows down to f/6.3 at 600mm. Thus the 150-600mm lens is anywhere from 2/3 to 1 1/3 stops 'slower' than the 24-105mm lens.
Aperture affects two things:
- How much light is allowed through the lens per unit of time. This affects the fastest shutter time we can use to get a desired exposure level.
- How much distance in front of and behind the point of focus appears acceptably sharp. This is what we call depth of field (DoF).
In both cases, a wider maximum aperture (lower f-number) is usually considered better. You can always stop down an f/2 lens to f/8 if you desire more depth of field. But you can't open up an f/6.3 lens to f/4 if you need a faster shutter speed.
How this applies to photographing wildlife, particularly at longer focal lengths, is significant. This is especially the case when photographing them in the early morning hours or late afternoon hours when they are most active. The dimmer morning/evening light combined with the movement of the subjects create the need for wider apertures in order to enable shorter shutter times (faster shutter speeds) without raising the ISO too high.
The sensor size is also significant with regard to how much total light is collected at a specific aperture and the depth of field rendered for a specific aperture. Larger sensors collect more light from a wider area for the same scene and exposure settings. In general, if both cameras have sensors with the same type and generation of technology, the FF camera can usually shoot at one stop higher ISO than an APS-C camera and get about the same signal-to-noise ratio (if we can raise the ISO one stop, it allows us to use a shutter time half as long, i.e one stop 'faster', to get the same exposure level). Larger sensors also give shallower DoF for the same focal length, aperture, and shooting distance. (Keep in mind that any cropping done after the fact has the same effect on DoF as shooting with a smaller sensor that would have given the same FOV as the crop would have given.)
With a 150-600mm lens, shooting technique must enter into the discussion as well. Even on a FF camera, at 600mm much more care must be taken to prevent camera movement from affecting the photographs taken. This is magnified even further when such a lens is used on an APS-C crop body. The narrower angles of view mean that the same amount of camera movement creates more blur than it would with a shorter focal length lens that gives a wider AoV. The narrower AoVs also mean the same amount of subject motion will cross a wider area of the frame (if the subject is at the same distance from the camera). Shorter shutter times can help, but support from a tripod or monopod is almost always required at focal lengths in the 300mm+ 'super telephoto' range. If one is tracking wildlife in motion at very long focal lengths, a gimbal mount that can support the weight of such lenses can be rather expensive.
Most of the seasoned sports photographers I know (some of whom have been published in major publications such as SI, ESPN The Magazine, the NY Times, etc.) rarely if ever shoot above 200-300mm handheld. You can get good results handheld past 300mm with a FF sensor, but rare is the photographer that can get great results at those focal lengths without more stable support. Monopods are very popular with sports shooters for 300mm+ lenses (on full frame bodies).
The final major consideration when shooting wildlife with these two bodies and these two lenses is the difference between each camera's non-sesnor related features: the AF system and handling speed.
There's not a lot of difference in the number of AF points, but the 6D's centerpoint is a little better in low light than the 650D's. My experience has been that Canon's FF cameras focus more consistently accurate from shot-to-shot than their APS-C bodies with the same generation AF systems. For example, the FF 5D Mark III has the same basic AF system as the newer 7D Mark II. I've found that the FF 5D Mark III gives me a higher 'keeper' ratio with regard to nailed/missed AF than the 7D Mark II does (they're both very good - but the 5DIII is better). I would expect the same difference between the 6D and 650D.
One difference that can become more significant with longer focal lengths combined with wider apertures is the ability to do Autofocus Micro Adjustment (AFMA). The 6D can, the 650D can't. At f/5-6.3, though, the increased DoF compared to using a wider aperture lens makes it less critical. In the case of your Sigma 150-600mm lens, if it is one of the new 'Global Vision' series of Art, Sports, and Contemporary lens lines, you can adjust the lens directly via the Sigma USB Dock. It's a fairly technical undertaking that can be a time consuming process, but it's worth the trouble to calibrate the lens and camera to work together.
In terms of handling speed the 6D is quite a step up from the 650D. Although it can only shoot at 4.5 frames per second as compared to 5 fps for the 650D, it can maintain that rate for much longer. The 650D bogs down after only 6 RAW or 22 JPEG images. With a fast UHS-1 SD card the 6D can shoot 17 Raw files or over 1,000 JPEGs before it fills the memory buffer. The additional controls on the 6D also allow faster changes of settings without having to press multiple buttons or dig into the menu. This is important when you want to be able to make those changes quickly without taking your eye from the viewfinder. To take advantage of it you will need to practice and become very familiar with using the controls with your eye at the viewfinder.
What works best for you and what you are shooting will have to take all of these factors into effect. Luckily, you don't have to make a fast decision and stick with it forever. Since you own both lenses and both cameras, you can try both combinations to see what gives you the best results.
Based on your own statement that you are a newbie, my personal recommendation would be to start out using the EF 24-105mm f/4 L IS on the 650D and learn to use the 150-600mm f/5-6.3 with the 6D. I'm not usually a fan of putting the EF 24-105mm on a crop body because it limits the wide angle end too much, but if you are shooting mainly wildlife it's probably wide enough for most of the other things you'll want to do while you are out shooting wildlife. For other situations where you need a wider AoV you can put the 24-105mm on the 6D.
Once you're comfortable shooting at 600mm with the 6D and have learned how to control the longer focal length and deal with the narrower aperture you can swap things around and try that. If you find that you want to keep the 150-600mm on the 6D and the 24-105 isn't wide enough for the 650D, maybe consider trading or selling the 24-105mm for something like the EF-S 17-55mm f/2.8 IS.
From a comment made by the OP to this answer:
When I travel I tend to only take the 24-105 with me, this question was specifically for a safari trip I'm doing next year where I will need both lenses and we will be in a vehicle. I will practice with both lenses on both bodies before we leave though!
The specific requirements of a safari adds a few unique twists as well.
- You'll be shooting most of your wildlife from a vehicle which may or may not be moving when you are shooting. A tripod may or may not be practical or even possible. Be sure to take a good, heavy duty monopod with a quick detach type head for attaching your big lens. Bean bags have their uses and are great for static animals, but a monopod allows one to pan a big lens more smoothly to follow running wildlife.
- At times the need for reach trumps everything else when shooting wildlife encountered on a safari type tour. It really does depend on how close your guides can get you to your subjects and how close your subjects are willing to allow you to come. Some tours are so regular that the animals are very used to the presence of the vehicles and humans and may even approach the caravan. Other tours can take a more arms-length approach to getting close to the wildlife and the animals are not conditioned to be approached as closely and will be more skittish.
- You don't want that large gap between a 23°AoV (105mm on FF) and a 10° AoV (150mm on a 1.6X crop) if the animals are approaching close to your vehicles or allowing the vehicles to approach very close to them. For reasonably close work of medium to large animals within a few dozen yards, between 10-20° AoV is where you'll want to be able to shoot. On the other hand, if the animals are never closer than a few hundred yards, you're going to want as much focal length as you can use effectively. The difference between 600mm on the 6D (≈4° AoV) and 600mm on the 650D (≈2.5° AoV) will be a major one if your shooting technique can get the most out of the lens at the narrower angles of view.
- A 'once in a lifetime' trip (which is what a safari is for many of us) is not the time to learn how to shoot with tools such as a 150-600mm lens, particularly one mounted on an APS-C body. Find similar situations close to home that you can practice with. Although you might not be excited about taking photos of run of the mill farm animals, if you practice shooting horses, cows, goats, pigs etc. from similar distances and shooting platforms you will be more prepared when you get those limited opportunities with more exotic subjects on your trip. If there are farms with livestock in your area, contact some of them and see if they would let you practice using their animals, even if you might only be shooting from the side of public roads that border their property. A local farmer's co-op might be a place to start to find someone to ask.
Full Disclosure: I often shoot with two bodies. My default setup when I'm not in extremely low light is an EF 24-105mm f/4 L IS on a FF body and a 70-200mm f/2.8 L IS II on an APS-C body. There's a big difference between 70-200 and 150-600mm, though. Even on the crop body, the 70-200 acts like a 115-320mm lens would on a FF body. That segues nicely from the 24-105mm on the FF camera.
I love the 24-105mm on a FF body because of the useful angles of view it provides and the pounding and punishment it can take and it still keeps on going. But if I need absolute image quality, I'm reaching for a better (usually prime) lens. When my primary body was an APS-C camera I used a 17-50mm f/2.8 as my main lens. Although I didn't own the 24-105mm at the time, it would not have been wide enough for my needed usage on a crop body.