Cartridges need to be compatible with the effective mass, height range and counter weight of the tonearm. The height of the tonearm should allow the dust cover to close (if there is one) and facilitate the vertical tracking angle.
The phono stage needs to be suitable for the cartridge's output and electrical parameters.
Moving Magnet (MM) cartridges have higher outputs and thus less gain is needed from the phono stage. Bass response and R/L channel separation can be good, but the mechanics of moving a higher mass limits high frequency tracing and decreases the threshold for mistracking. They are physically a bit more robust and some brands offer replacement styli without having replace the whole cartridge - or realign it.
Moving Coil (MC) cartridges have a lower effective tip mass with a thinner shank and can trace higher frequencies more easily and better reproduce transients. The are more delicate and performance may drift if critical suspension elements change their physical characteristics over time.
A high performing tonearm and table is required to achieve the sonic potential of either type. Moving Coil (MC) cartridges require a more elaborate - and therefore more expensive - phono stage for low noise and good dynamic range.
Cartridges are alignable in all three X, Y and Z rotation axes and can be translated horizontally in the head shell of the tonearm and vertically translatable by moving the tonearm up or down. Sound quality will be more affected when a cartridge is less than optimally aligned especially with more elaborate stylii profiles.
While frequency response, channel separation and output are all affected by the cartridge loading adjustments of the phonostage as well as alignment, mistracking of the more heavily modulated passages, particularly towards the end of the side, spoils the listening experience and may cause groove wall damage over a large section of the groove wall resulting in a permanent loss of detail.
You can install and reasonably set up a cartridge with a few basic tools and some cartridges are far more forgiving than others. Bear in mind that all alignments are a compromise and that sound quality will vary from record to record based on the cutting tool used and the record's thickness.
While most manufacturers provide the tracking weight range, bias compensation is not usually specified (and actually the amount required varies across the record!!)
Hard shellac 78 rpm records would wear out crude inexpensive steel styli after a small number of plays. Today's laser cut diamond styli last a lot long longer and depending on the geometry and degree of fine polishing a stylus will typically last for between 1000 to 2000 record plays.
When a cartridge is correctly aligned the gradual loss of high frequencies as the stylus ages is more likely due to a worn stylus than a worn groove. Changing to a stylus with a broader area of contact may recover high frequencies from a section of the groove wall that has not been worn in well-played records.
The diamond tip is a big cost factor in the price of a cartridge. Inexpensive ones use bonded styli - a smaller cap of diamond glued onto a stub. Premium cartridges use whole "nude" diamonds that can be machined into the more complex geometries that provide the broader groove contact area. The hardness quality and degree of polish is also a factor. Stronger magnetic materials are desirable but add to cost as does hand-assembly.