How important are the Speaker Cabinets?

They are an integral part of the overall design so they are important. The overall design matters more than any particular cabinet shape or size. Generally, they should be very rigid, which infers greater weight. Larger cabinets cost more to build and maintain high rigidity compared to smaller.

One good example of a speaker cabinet built to the nth degree is probably that of the Magico mini, which is a rather small 2-way midsize monitor with integrated stands that cost over $30000. They can command that price not only due to good sound in a relatively compact package but build quality of the highest degree.

The volume of air inside (and porting) as well as stuffing will create a resonant system with the woofer. Cabinet volume and porting are matched to a particular driver. This is extremely important for the woofer.

Furthermore the edges and shape of the speaker may influence the edge diffraction of frequencies above about 500 Hz.

Furthermore cabinet resonances may color the sound and increase distortion.

the best cabinets are closed perfect cubes, no prorts or internal structurs. You can just stack as many as you want on atop the other. T walls must be free to resonate at 500 cylces thus use a thin copmpliant material. Make sure that no one slips any damping materials into the box .
The only downside is that they are a bit skittishhis cab be aproblem as some of the best made of really thin brittle material can break if it bunces off it's perch to the florr. If made right it will shatter. They make a really loud buzz /hum at 500 Hz.The family often gathers around to witness this like the first planes to break the spund barrier.
The pretyentious speaker makres use heavy damped non paralell walls to make them fancy. They add wieght with extra internal bracing -it's only purpose is to make the speaker real heavy and rthus feel like something substantial. That gives you the sense that they are a prestigious product.
Don't fall for that industry tomfoolry.

There are various ways to skin this cat.

Just as in turntable plinths, one school advocates the addition of mass, in the form of thicker cabinet walls and additional internal bracing, to make the cabinet as rigid as possible. All cabinets resonate, always at more than one frequency, and the goal of this design is to quiet those resonances as much as possible. The addition of mass, usually lowers the frequency of the resonanance and lowers the "Q".

Another design school advocates making the cabinet lighter. The obvious effect is that the resonances, though louder, will be higher in frequency. If they can be moved up in frequency, they are both less bothersome to the ear, and they tend to be of higher "Q". "Q", or quality factor has to do both with how wide the resonance is and how quickly it clears, or stops ringing. Even if the high frequency resonance is louder, if the ear is less sensitive to resonance in this range and the resonance stops sooner, it will have a different sound than the other example, above.

Material is an important consideration in the resonant structure of the cabinet, which is why some manufacturers use exotic materials, such as the aluminum cabinets of the Magicos referred to in the above post. And some use different materials in different parts of the cabinet to spread resonances and to add some degree of economy to the equation.

Although most speakers are made of MDF (medium density fiberboard) which is generally cheap and may be comparable to the cost of some drivers, the labor in constructing and veneering the cabinet is the more significant expense in the finished cost of most speaker cabinets. My guess - and it is just that - is that most cabinets well exceed, by serveral times, the cost of the drivers in them. And this is most certainly true of those with exotic materials and unusual construction techniques.

But in the end, knowing if the eggs are brown or white does not tell you a lot about the finished omlette. Listen without predjudice to various speakers and pick the one that most closely matches the paradigm of live music that exists within your mind.