I completely agree with the sentiments above - the primary difference determining the quality of the photo taken with an iPhone 7 Plus and another taken with a Canon SX720 will be the skill level, imagination, and dedication of the person pushing the button. However, there are measurable differences due to sensor size so I'll focus on those.
Let's break down the effects of a larger sensor into four categories:
- Size of the individual photo sites on the sensor
- Lens mechanics
- Photographer ergonomics and other secondary effects
- Post-procesing capabilities in-camera
Larger Photo Sites
A larger sensor will generally - although not necessarily! - have larger photosites, which means more photons are hitting the sensor for each individual pixel (in the bayer array). An APS-C camera has a diagonal which is 1/1.6th the length of a full frame sensor, which means an area which is 1/2.56th or about 0.4 of an area. Thus, we would expect a full frame sensor with the same pixel density as a APS-C crop to have 2.56x as many megapixels. So, the pixel density of a 22MP full frame sensor is roughly equivalent to an 8.6MP APS-C camera. Conversely, the full-frame equivalent to a 24MP APS-C (ex, Canon 80D) would be 61.44MP.
Obviously talking across different generations of sensors this doesn't hold true. Each generation the interstitial distances between photosites on the sensors have decreased, the base sensitivity of the sensors has increased, and the various sources of noise have decreased.
The size of the photo sensor allows, with an identical lens, more (or less) light to hit the sensor in a given amount of time. If more light is hitting a photo site then you have a more accurate representation of the brightness at that photosite; if you are getting only 10 photons hitting your photosite then if one or two do not you have a 10-20% darker photosite reading than you should; if 100 photons are hitting your photosite then if one or two do not you have a 1-2% darker photosite. At the same time, the sensor is an electronic circuit which is subject to electronic noise which can increase/decrease the reading of each of these photosites by a very small amount, which amount is the same if the "signal" is 1 as if the "signal" is 100.
These readings then get "amplified" in circuitry to the point where they can be discernable when output, so another way of saying the same thing is the more "gain" you have to apply (represented by a higher ISO rating) the more obvious the noise will be. You will "need" a higher ISO (more gain) to get a viable picture in a smaller sensor camera (or you can apply gain afterwards out of camera, but then the low-signal image is subjected to more sources of noise before it is output, so generally getting the ISO right "in camera" is key; there are some Sony sensors [Nikon bodies] which are said to be "ISO-invariant" within certain ISO ranges, which essentially means that they have controlled pipeline electronic noise introductions so well that amplifying the signal all the way in Lightroom is essentially equivalent to amplifying it in-sensor with ISO sensitivity).
Now, looking at the 20MP in 1/2.3" (Canon SX720) versus 12MP in 1/3" (Apple iPhone 7 Plus main camera). The crop factor there is 1.3x, which yields 1.7x in area. The number of pixels is 1.67x, which to my eye is vanishingly close (within the limits of measurement) identical. So, the Canon's sensor has no photosite size advantage despite being larger. It just has more pixels around the edges.
There was an important caveat above, which is that all of that is true with an identical lens. For instance, take a Canon 5DmkIV and a Canon 80D and put any EF lens on them both, and the math around how much light gets to each photosite is absolutely true.
However, if you had a "larger" front element lens, that lens will have more "light gathering" power. This is to the square of the diameter of the front element (assuming no additional losses in-lens), so a lens 1.4x as large at the front will gather 2x the light to start transmitting down to the sensor. Think of a simple magnifying lens and lighting kindle on fire; if you have a little small magnifying glass, you might get the kindling just warm, but if your magnifying glass is larger you might get it hot enough to catch fire.
However, such a lens will also need to do much more "bending" of that light to get it projected down to the sensor size. And this is the crux of the issue: bending light introduces inefficiencies where light is lost (especially around the edges of the image circle), and the only way to combat those inefficiencies is to use more expensive lens systems with heavier elements, or to extend the length of the lens system.
If that "bending" doesn't happen perfectly, then light which should end up in one photosite instead hits another photosite, leading to poor focus.
The end result here is that with a different "lens mount" specification, you can drastically change the image quality possible with reasonably-constructed lenses. Talking specifically about phones and point-and-shoot lenses, there is no "lens mount" since they are designed for a single lens system, but the same principles apply.
Okay, all that said, you are comparing a fixed "prime" lens on an iPhone 7 to a "40x zoom" lens permanently attached to your SX720. A zoom lens almost by definition will have lower image quality than a prime lens. So, if you are trying to take an identical picture with the Canon as with the iPhone, you are likely to be disappointed because the iPhone's prime lens will likely far outshine the zoom lens zoomed out to ~20mm equivalent.
A great photographer can get a great photograph with a toy camera, and a bad photographer will not get much benefit from the best camera. Good ergonomics help a photographer be better with the camera - how sturdy is it to hold the camera, how easily can the settings be manipulated with good feedback to the results, how likely are they to have the camera with them or pull it out of its bag in the first place? Ergonomics won't rescue a miserable camera nor a miserable photographer, but it can make the photographer make better use of the equipment.
Generally, the layout of buttons on a good point-and-shoot, especially a larger one like the SX720 will help the photographer a lot compared to the flat rectangle of any phone. But, with a phone's touch screen a lot more control can be more optimally organized (again, though, point-and-shoot cameras are tending towards touch screen controls in addition to basic button controls as well).
A larger sensor requires a larger lens to feed it light. Lenses are heavy. If you were talking about a DSLR, then you also have the heft and bulk of the flipping mirror, pentaprism, and lens mounting system. If you were talking about an interchangeable lens mirrorless, strike the mirror and pentaprism, but all the rest is still needed. It is a lot to carry around, so generally such systems don't get carried around except when there is known to be a need for them.
Point and shoots are obviously on a different scale than a DSLR, but given that almost all of us would scarcely imagine leaving home without our phones, the bulk of a point and shoot (and the SX720 is far from svelte) will keep you from taking it just the same as the DSLR. Doesn't mean it is a worse camera, just that it is one you can't rely on being there all the time.
Most of all, if you don't have your camera with you, the photos that day will completely suck. But, if you have multiple cameras to choose from, choose the one which gets the best results for you. When I'm going out somewhere where I expect to be taking pictures, I'll take my DSLR with me. That won't stop me, however, from whipping out my iPhone to take pictures where i didn't expect.
Finally, back to the zoom lens which I touched on above, this section is where the SX720's 40x zoom shines. You will get a much better picture of a lion with the SX720 than you are likely to survive getting with the iPhone 7 Plus. No question. Zoom lenses are popular because they allow us to take pictures much more easily, frame them far more precisely (rather than cropping after the fact, which essentially further decreases the effective sensor size), and stay alive to complete the post-processing. That is a major ergonomic win.
Since you are looking at an Apple iPhone as one example, you need to take into account the very advanced logic Apple puts into its phone (and Apple is not alone here obviously, but generally the phone makers have a lot heavier reliance on post-processing computing power than any of the point-and-shoot manufacturers, because that processor is already there for them to use).
With a point-and-shoot the post-sensor electronics are generally minimal, just enough to apply the needed effects like white balance and noise reduction before outputting a JPEG image. Canon's Digic line, for instance, applies rote processing to each photo but has a primary characteristic of how quickly it can commit the image to storage (and with as little battery power consumption as possible). With an iPhone, the camera frameworks (software) are taking a lot more into account to improve your photographs. The processor is one more powerful than personal computers a decade ago, and the battery is sized to keep a large screen and wireless transmitters going for the day so processing a photograph is just a drop in that bucket. So, the iPhone can detect faces more accurately and micro-tone so that the skin tones it finds are kept natural while retaining the blues of the sky and greens of the grass, etc.
Now, this all seems nearly sacrilege to a dedicated RAW photographer - I want as little done to the sensor data as possible until I get it into Lightroom, so I can make and fine-tune those decisions to my own eye. But if you are ending up with a fully-baked JPEG anyway, a camera with a strong software processing unit and flow will generally give astoundingly better results than you would expect with the hardware being used.
They both look to be excellent cameras. I wouldn't worry about which will give you better photos in and of themselves, but if I had to choose between having one or the other, I'd choose the one I am more likely to be carrying when I need it.