Not quite sure exactly what you are asking (recording, live sound?) but as a guitar player who also is responsible for the PA for my band, and as a songwriter with an in- home recording studio - I have studied on this and experimented over the years.
Here are some thoughts to consider. Different speakers are designed differently to do different things. A guitar cabinet or bass cabinet has a speaker designed to reproduce the sounds of the instrument (within a specific sonic frequency) in a manner that the guitar player (or bass player) can hear. In many live applications the guitar cab is mic'ed and the sound is fed to the main and monitor speakers through one or more mixers - so it is sent to the audience through the main PA speakers and the musicians on stage through floor or flying monitor speakers.
Guitar cabinet speakers are not designed to direct sound to a large audience and tend to be very directional. So they would sound different depending on the angle of the speaker to the listeners ears, and of course whether the cab was facing or behind the listener. That's why in live settings with higher volumes for larger audience, the musicians will usually be listening to their instruments through the monitors, not the amps (even though they may also hear some of what is coming out of the amp).
Most of the time, guitar cabinets designed strictly for guitar (or bass), have one or more speakers (usually identical), that are optimized to produce the frequencies for the intended instrument - whether bass guitar or standard guitar. Some acoustic amps with microphone inputs may also have two different sized speakers in order to cover the more varied range of vocals in addition to guitar.
Coaxial speakers were originally designed primarily to save space. By positioning a higher frequency driver inside the cone of a lower frequency driver, two drivers can occupy the same space as one. A true coaxial speaker is actually two separate speakers (drivers) mounted concentrically, each with their own terminals and usually with a built in crossover.
Recent developments in the design of high fidelity coaxial speakers have been geared more towards taming inconsistencies in the off axis sound reflections that are characteristic of a vertically stacked driver orientation. By putting both drivers in concentric relation to one another, the sound radiates from the same point in space and will reflect off of horizontal planes (walls) AND vertical planes (such as a ceiling) at the same angle.
PA (Public Address) speakers are designed to deliver sound to a large audience over a wide area. The cabinets and speakers ideally should be designed for as wide a dispersion pattern as possible in order to broadcast clear sound to a wider area. Typically at least two PA speakers will be used to broadcast sound to a wider area. In larger venues many more may be used. The placement and orientation of these PA speakers will depend on the acoustic properties of the venue (if indoors) and where the audience is in relation to the speakers.
A typical PA speaker for most popular music will have two speakers (known as drivers) in the enclosure - a low frequency driver and a high frequency driver. You can find them with a low, mid and high frequency driver known as a 3 way speaker.
The PA speaker must cover the entire frequency spectrum as they will be tasked to deliver not only the sound of the instruments, but the vocals as well, to the audience. For bass heavy music you will often have one or more sub-woofers incorporated in the the PA speaker array to handle the lowest frequencies (bass guitar and kick drum).
The low frequencies will be directed to the sub woofers through a crossover either in the speakers themselves or in the mixer, depending on the equipment used. The cabinet of a sub-woofer (as well as many bass guitar cabinets) are designed to optimize the low frequencies. The cabinets are engineered to reflect and radiate the sound waves coming from a large speaker cone moving over a larger area at a slower rate. So the cabinet design itself will have a significant impact on the way these low frequency speakers sound.
PA speakers are constructed to withstand higher wattage amplifiers required to render distortion free sound to a larger audience. Lower wattage amplification may not drive them sufficiently to sound good so the same type speaker would sound bad in a 100 watt guitar amp - but the speaker cone in the guitar amp would be reduced to shreds as soon as a 800 watt PA amp sent an electrical signal to the voice coil.
Some PA speaker systems feature an articulated array of speakers angled to point in different directions to widen the dispersion pattern. In addition to the articulated speakers, the Bose L-1 and L-1 compact speakers for small to medium venues, also feature a line array system that uses the physics of the sonic wave form to allow the sound waves from multiple speakers to push each other along, thereby reaching a greater distance with lower volume. In large venues, this line array phenomena is used by arranging multiple PA speakers in a vertical array - either stacked on the floor or stage or flying overhead from trusses.
Because of advancement in amplification technology allowing for high powered amplifiers that are extremely light weight, there is a trend in PA speaker design towards what are known as powered PA speakers. These speaker cabinets have a built in amplifier just for the drivers inside that one cabinet - and tend to offer more headroom and higher volume before distortion than many of the passive speaker systems offer.
In live settings there is one other type speaker that you will see. The stage monitors are designed to allow the musicians to hear what they are playing. They are very similar in design to PA speakers in that they must deliver the entire frequency range, but are usually smaller and often wedge shaped so they will angle up towards the musician's ears when sitting on stage.
For studio use, the speakers are designed for close proximity monitoring (listening) in a very quiet room with controlled sonic properties. They don't require as much amplification as a PA speaker and the design can focus more on optimal sound quality than ability to withstand high wattage amplification. Studio monitors will generally be two or three way with possibly a low and mid and high frequency driver to cover then entire sonic frequency range.
But while studio monitors sound great in that very controlled listening environment, they would not be effective at all for live sound reinforcement.
It would be great if we could get the same high fidelity sound that studio monitors are capable of producing in a live music sound reinforcement application. But even if we could, the varying demands (indoors, outdoors, acoustic properties of the room, ambient noise) of the various settings where live music is played, would have as much of an impact on what the listener perceived than the speaker design itself.
But I have been really impressed with the sound of some of the newer powered PA speakers.
Hope this will help shed some light on speaker design properties for different applications.