It depends on what the camera lets you do. More and more new cameras / firmware updates allow you to shoot in Aperture priority with auto-ISO with settings for both minimum/max ISO AND minimum shutter speed. With that logic, you're really controlling both aperture and shutter speed - the camera is actually changing shutter speed as needed, but only according to the very specific instructions you give it. For example, lets say you're shooting action and want to maintain a minimum shutter speed of 1/500. You can then set the camera with a max ISO of 3200 or 6400 or whatever you're comfortable with as a maximum ISO, and a minimum shutter speed of 1/500. Then you set your aperture and shoot. The camera will maintain the lowest ISO it can while maintaining your minimum shutter speed of 1/500. In really bright light, it will maintain base ISO and will increase the shutter speed as much as needed for proper exposure. As the light decreases, it will increase the ISO as high as needed (up to your designated maximum) to maintain your minimum shutter speed. If there's too little light to maintain your minimum shutter speed even at your highest allowed ISO, ONLY THEN will the minimum shutter speed start to drop below your designated minimum shutter speed as needed. This is exactly the same logic and progression I've used when setting these parameters manually. To me, this is the best possible setup for auto-ISO except in the case where you purposely want a slow shutter speed to induce motion blur or something - at which point THEN you can easily switch to manual or shutter priority mode.
Using Manual mode with auto-ISO allows you to use the same logic but it's far more labor intensive. It's really not very different than shooting in Aperture priority with manual ISO. Shooting in either of these modes (Manual with auto ISO or Aperture priority with manual ISO) requires that you set your primary parameter (let's say it's aperture) and then adjust the secondary parameter, but while constantly monitoring the third (automatic) parameter to make sure it's within a reasonable range. In aperture priority and manual ISO, you set the ISO, but you're constantly monitoring the shutter speed to make sure it's not getting too high or low, and adjusting the ISO manually as you need to. In manual mode with auto ISO, you set the shutter speed but have to constantly monitor the ISO to make sure it's not getting too high or low, and adjusting the shutter speed as you need to.
Using Aperture Priority with auto-ISO and designating a minimum shutter speed will do the same thing, but it's balancing the second and third controls the same way you would, but doing it for you. For example, using manual mode with auto ISO on a bright day, you may not have any problem staying at base ISO in your chosen aperture, but you'd have to be raising and lowering the shutter speed if you haven't set it fast enough for conditions. With auto ISO and a minimum shutter speed, the camera will go as fast as it needs to without you having to do anything, but you know you're also protected if you or your subject is moving through shadows and it needs to drop the shutter speed to your minimum and raise the ISO somewhat. Similarly, as light gets lower, in manual with auto ISO, you'd have to constantly lower the shutter speed as needed to keep the ISO in a reasonable range. In aperture priority with auto ISO and minimum shutter speed, it'll find that balance for you at the point that maintains your minimum shutter speed with the lowest possible ISO.
So, on balance, I don't really find using manual mode with auto ISO much different in practice than shooting in aperture priority with manual ISO. Either way, you still have to set two of three parameters based on careful monitoring of the third, automatically adjusted parameter. I used to go back and forth between these two modes with my Ricoh GXR and couldn't find a difference in how I shot. If anything, I preferred aperture priority with manual ISO because shutter speeds had a much wider range of possible values to be adjusted automatically than ISO, so I was better off changing ISO as needed and letting the shutter speed float than vice versa.
The only problem with using Aperture priority with auto ISO and minimum shutter speed, as I've outlined above, is that not all cameras do it, or do it well enough. Samsung and Nikon do it incredibly well, and Sony's newest models (A7R II, RX100 IV, and RX10 II) also adopted this industry standard that Nikon and Samsung have been using, but earlier models didn't offer anything like it and Sony doesn't appear to do firmware updates. Leica's newer models have this feature. Fuji's higher end models offer this now, with the caveat that you can't set the minimum shutter speed faster than 1/500, which might not be enough for some serious action shooting. Ricoh has the same problem to an even greater degree with a fastest minimum shutter speed of 1/250. Canon offers it through an auto ISO feature they call "rate of change", but they offer far less specific control than Sony, Nikon, Fuji, or Samsung. Panasonic doesn't do it at all. Olympus doesn't either, but there's a workaround that allows you to essentially do the same thing up to a max shutter speed of 1/320. I'm not sure about Pentax or Simga.
Auto ISO has become an overwhelmingly powerful tool with today's high ISO and high DR sensors. When paired with a good shutter speed control within the auto-ISO menu, it's revolutionized how a lot of people shoot. Without that feature, however, it's just a matter of choosing your poison in terms of what you'd rather control and what you'd rather the camera controls.
-Ray