Transistors, power transistors in particular, have a non-linear capacitive aspect that exists in the junctions of the device. It is multiplied with current. This contributes to odd-ordered harmonic generation. Since our ears use odd orders to determine sound pressure, this has a big effect on the resulting sound- it will sound louder and brighter even if odd ordered harmonic generation is only 100ths of a percent. Our ears are tuned to listen for them- we are more sensitive to their presence than we are human vocal frequencies.
The use of feedback in audio amplifiers causes the noise floor of the amp to change. Instead of mere noise, from active and passive components, the noise floor becomes a harmonic and inharmonic noise floor, much of it the result if intermodulation at the feedback node. The ear can hear 20 db into a natural noise floor despite the masking characteristic, but cannot penetrate the harmonic noise floor of an amp with feedback. This means that such an amp will be inherently lacking low level detail, sheerly out of the ear's masking rule.
The problem is that with the odd-ordered harmonic generation, people will be screaming at you to turn the transistor amp down (if you don't have the sense to do it yourself). Tube amps tend (this is a generalization of course) to have a larger percentage of *usable* power and as clipping is approached, will do so gracefully enough that the ear is fooled into thinking that the sound is louder than it really is (due to the presence of odd ordered harmonics) but the harmonic generation may not be so bad that one wants to turn it down. Electric guitar players take advantage of this all the time (they use tube amps for the most part).
Transistors get called 'bright', 'harsh' etc. on account of the odd ordered harmonic distortions they make (although tubes can certainly be guilty of this too). Since our ears are tuned to these harmonics, it should not be any surprise that one *can indeed* have certain expectations- usually based on experience.
Tubes have all the bandwidth you could ever want. An analog color TV needs video circuits with MHz worth of bandwidth- and would not have been possible if tubes did not have bandwidth. More to the point, bandwidth is a product of the designer's intent, not the device. You can have output transformers that have MHz bandwidth- the ZERO is in fact such an example. For practical reasons excessive bandwidth can be headache as RFI can be a problem. We once built an amplifier that was effective as an RF amplifier at 30MHz. We had to do a lot to keep RF out of the amp, else it fried tweeters pretty quick for 'no good reason'
Our MA-3 can make full power at 100KHz. So when you talk about bandwidth, its probably more accurate to talk about transformers being the possible 'limitation' rather than tubes generically.
This is not entirely true. For example, there are no loudspeakers made that are 8 ohms and need more than 20:1 for a damping factor. Some speakers need far less- there are even speakers that are over-damped if the damping factor exceeds 1:1.
IOW you cannot make such a generalization- this is because even though the industry wanted tubes to go obsolete 50 years ago, they failed to do so. As a result there are plenty of speakers around that need lower damping factors in order to sound right. The very first sealed box speaker, the Acoustic Research AR-1, is such an example. It was designed for an amplifier with an output impedance of 7 ohms.
The problem is that the industry saw an opportunity to make money. When transistors started to become practical, the industry realized that with feedback they could linearize the amp and sell it for 90% of the cost of a tube amp of the same power, but with only 10% of the parts cost. This was a big incentive! Since such amplifier designs rapidly took on the ability to double power as impedance was halved, the idea that an amplifier should be a Voltage Source emerged. But tubes don't double power in that way, yet have always had their adherents. As a result there are speakers that don't work with amps that don't double power, and there are speakers that don't work with amps that do. for more information see
http://www.atma-sphere.com/Resources/Paradigms_in_Amplifier_Design.php
If you mix the two technologies, tonal abberations emerge. IOW, there is no blanket statement like the one immedately above- the correct statement is that amps and speakers have to be matched together for best results and there is no way around that. If you have chosen a speaker that only works with a voltage source amp, then expect to be painted into the transistor corner. The converse is also true- there are many speakers that exhibit tonal abberation if transistors are used. Quite literally in some cases the crossovers don't even use the same rules! You have to match amps and speakers, and the spec sheets aren't going to tell you a lot about this as its sort of inconvenient truth that the industry likes to sweep under the carpet.