In "the olden days" everyone had a point and shoot film camera. These typically had a non-removable zoom lens with perhaps a 35-70mm or even 28-70mm zoom range, small size, and with a small rangefinder-style viewfinder. Everything was automatic, with some manual control available.
When the world went digital there was a bewildering variety of tiny point and shoot digital cameras launched, mostly pretty much identical, with tiny size and usually no viewfinder. Image quality was fine for holiday snaps but the tiny sensor wasn't up to enlargements or challenging lighting conditions.
Then smartphones appeared and compact point and shoots have by now almost gone. The world seems to have split between big serious cameras (DSLRs and mirrorless ILCs) and ever more capable smartphone cameras.
But you can still get an in-between option and for some it might make sense. Today's fixed lens point and shoot still has the 28-70mm zoom range and fully automatic functionality but most people buying these things want a little more manual and creative control (otherwise they'd just be using their smartphones) so you get full manual control and (usually) Raw file recording. Typical examples are the Sony RX100, Canon Powershot G5, or Panasonic DC-LX100.
Think of a typical mirrorless ILC and get rid of the "IL" bit - i.e. the lens is fixed to the camera. A limitation in some ways but what about the advantages?
The camera designer can optimise the whole design around a single lens and sensor (you hope).
The lens and body can be much more compact when the lens isn't removable.
The big negative is the usually quite small sensor that most of these cameras have, limiting enlargement capability and reducing dynamic range (and increasing noise).
So why would you buy one of these things instead of going for a "proper" DSLR or mirrorless ILC, or simply taking your smartphone everywhere with you? Good question. I can't give you a good reason but as I've stressed in all these posts it would come down to handling, for me. I hate taking pictures with smartphones. Hate it. A compact camera in this category is designed to handle like a camera, so pick one up and see how it feels in the hand.
Crucial to me would be the viewfinder. Assuming it has one. Because cameras in this category still include the tiny viewfinder-less style that became popular in the late 2000s but also include the larger more serious looking styles that have electronic viewfinders and a body style like a very small SLR or rangefinder.
If you're going to look at one of these, consider:
Does it fit in your pocket? Is it sleek enough to put in your pocket and get out again without snagging?
Is it really compact? I've seen some cameras in this category that are larger than some DSLRs and I can't see the point.
Is it responsive in use? Cheaper prices can sometimes mean cheaper components and can lead to sluggish performance.
Does it have the kind of smart and AI features you'd expect of a smartphone nowadays? I'm thinking of automatic panorama stitching, automatic exposure blending, face and smile detection, top-notch auto-focus, background blur, night mode, etc. If you're going for a compact automatic camera you might as well have all the automatic features than modern computation can provide.
Also in this category, but with a distinct identity and purpose, are the super-zoom cameras. The Nikon P1000 takes this to the extreme. These cameras have all the bells and whistles but with a very big zoom range. The Nikon has the ridiculous range of 24-2000mm (in full frame equivalent terms) which is such a lot of zoom that I just can't see it being usable and the end result is one HUGE camera.
But the superzoom at least has a distinct purpose and reason for buying it - you just can't get this level of telephoto lens for less than many thousands of pounds with a DSLR or mirrorless ILC.
If I ever considered buying one of these cameras (super-zoom excluded, which I can't imagine ever wanting) it would all come down to size and handling. Would it fit in a pocket, is it always available, and is it fun to use? Can I walk round with it in my hand all day without anyone noticing? Does it have a lens cap you need to take off or is the lens cap integral and automatic? What's the viewfinder like to use? Does the rear screen tilt in useful ways?
I can definitely see myself owning such a thing for travel, or discrete city shooting or simply to put me in a different frame of mind and to get me out of a rut. My recommendation, as it has been in each of the other posts, is to get down to a real camera shop and try one out in the hand.