Cost of the material
With little exception, ceramic tile is going to be the least expensive of the three, followed by porcelain and then natural stone

Chipping
Porcelain relatively speaking is the hardest of the three materials. Stone can be softer, however the nature of ceramic makes it the easiest to chip. Stone is colored through so a chip isn't as noticeable, with ceramic the chip will usually stand out like a beacon. Porcelain has a more expensive family of through-body porcelain, where the color is contain through the body of the porcelain, however the variety of patterns and colors is limited

Installation
Without a doubt ceramic tile is the least expensive to install, it has great forgivingness and can be installed with the simplest of thinsets. Porcelain takes more care, requires more expensive thinsets, more expensive tools and much more careful installation due to the nature of the material and it's resistance to adhesion unless it is installed correctly. Stone is by far the most expensive to install, with additional labor required for pre-sealing of the stone if required, the thicker thinsets because of the irregularity of the material and the more expensive thinsets required. It's probably not common knowledge to homeowners but some marble will actually curl and warp if the wrong thinset is applied to it.

Looks or esthetic Value
Stone has a look of it's own, it is ageless and timeless and beautiful. When it comes to porcelain, the best porcelain is usually judged by how well it mimics natural stone. Ceramic when it tries to mimic a natural product fails miserably, ceramic has a look of it's own and it's best to use ceramic for what it is known for which is richer solid man-made colors that are not trying to look like a natural stone material.

Variety
This is an interesting criteria. Looking at stone you could say since it's a natural product that the variety is endless, however while there are endless variations of colors, veining and such there are definitely a finite amount of varieties of stone, from slate to marble to granite. The variations in the different families of stone, are what makes it so versatile. Porcelain we would consider the work horse of tile with a large amount of variety, but all trying to mimic natural stone. Ceramic while there is a multitude of 'artsy' one-off ceramics the selection of ceramic is pretty limited to mimicking the popular colors of the color palette.

Maintenance
Good quality ceramic floor tiles will not wear out easily. Porcelain can be considered as providing the same esthetics of stone without the problems. Stone's ability to be trouble free depends a lot upon the end users initial choices, honed will be more trouble free then polished, granite will be more trouble free then a soft marble. The maintenance issues of stone products are relative to the end user's expectations and understandings of the material and how it will be used.

Moisture Resistance
Stone can range from moisture resistant to moisture sensitive. Ceramic tile tends to have a moisture gain from between 3%-7% of it's weight. Porcelain by contrast is considered impervious to moisture, meaning it is limited to a moisture gain of no more than 1/2 percent of it's weight. This impervious nature is one of the factors that make it more expensive to install than ceramic tile, porcelain with hardly any pores  and thinsets work by bonding to pores makes porcelain installation a less forgiving process.

Refinishing
Not an often thought of factor and for many not an option. However, a stone floor can easily be reground in place,  honed or polished again and look brand new. A through body porcelain can go through this process also. Ceramic has no option.

Consistency
Ceramic and stone are on opposite ends of this scale. Stone is a natural product and variations in color, texture and appearance are what gives stone it's natural appeal. If you like stone, you better be a fan of natural variation. Porcelain will straddle this line depending upon the style of porcelain you are looking at. Some porcelain is made on purpose to have a lot of variation and some is made to have more consistency. Ceramic will be the most consistent of all. After all a box of 4 inch white ceramic tile is the pinnacle of consistency.

Quality Control
Stone definitely has the potential for the most problems in this regard. Stone isn't manufactured in a factory but quarried and resold, which always opens the door to problems of quality when new quarries are being bought from usually from the other side of the world.

Exterior Applications
Stone if often considered the superior material for outdoor applications, however certain grades of porcelain are just as advantageous if not more. Ceramics are not suitable.

Status
Yes, there are people who choose materials for many reasons and there is nothing wrong with it. Stone has no equal in this department. It is considered the finest material so it leads the pack. Stone is considered a one-of-a-kind material, no two projects will be identical, each will have it's own character and be uniquely different, exotic and beautiful, stone will age gracefully and change with time. No matter how close porcelain comes to mimicking stone, it is not ever going to be stone. The fact that people often choose porcelain based on how well it mimics stone should tell you that stone is the material used to judge all others.