I wish there were a scatter diagram one could put together of tube amp sound qualities.@jimbones The last thing the industry wants is for the buyer to know what the amps sound like :) They spend a lot of time trying to tell you that they don't sound like anything at all.
But there are differences and if you have enough information you can figure out what the sound will be to a certain degree.
So here we go:
In all cases keep in mind that this all about linearity.
There are differences in power tube types roughly split between triodes and everything else. Triodes are more linear and so will sound smoother and more detailed. But there is Ultra-linear operation (an output transformer thing), which if properly executed, results in a pentode or tetrode having triode linearity as well. The key in this latter bit is 'executed properly'. Many OPTs (OutPut Transformers) have the screen taps improperly set to avoid a patent infringement of long ago, and that's become a bit of a tradition- often designers don't question the taps.
You have single ended VS balanced and a combination of the two; the latter of which is most common. Single ended circuits have a quadratic non-linearity which causes them to make a fair amount of 2nd harmonic distortion. Balanced (often differential) circuits have a cubic non-linearity, which causes them to make a 3rd harmonic (in theory the even orders are cancelled throughout the circuit, resulting in dramatically less distortion as distortion is compounded less from stage to stage), with succeeding harmonics falling off at a faster rate as the order of the harmonic is increased. The ear treats the 2nd and 3rd the same- they contribute to the 'warmth' of tubes.
So an SET can sound quite warm and vivid because of its distortion. It can also sound dynamic on the same account if you are not careful about the speaker selection. A fully balanced amp will also have some warmth but will sound more transparent owing to reduced distortion. Because of the reduced distortion, it will also sound less 'loud' at higher volumes.
To give you an idea of how profound the difference is in distortion, almost any SET will produce 10% THD at full power which might be only 7 or 8 watts. A fully balanced amp using the same tube types might produce 1% THD at full power, and full power will be about 4x greater if the same power tubes are employed. So at 1 watt (keeping in mind the first watt is the most important) a balanced amp can have well over an order of magnitude less distortion. This is heard as clarity.
If the amp has single-ended input circuitry combined with a push-pull output, the two non-linearities are present and due to algebraic summing, the 5th harmonic becomes more prominent. Most tube amps fall into this category. The 5th isn't a good thing- this makes the amp a little less musical.
The class of operation can have an effect. Class A amps generally are more linear and so make less distortion. IOW they will be smoother and more detailed. This is not a hard and fast rule; if the designer is careful a class AB1 amp can have low distortion as well; the designer has to be more careful about preventing distortion from cropping up when the signal is crossing the 0 point between positive and negative.
Generally speaking, the more complex the circuit the greater the places things can go wrong such as the effect of additional coupling caps, loss of bandwidth, increased distortion and noise.
Finally feedback plays a role too. In tube amps feedback is useful in getting the amp to behave as a voltage source; in most amps this would be about 15 dB of feedback (feedback is the practice of taking some output signal and applying it to the input of the amp, out of phase with the input signal so it can act as a correction voltage). The problem here is that feedback in such an amount adds higher ordered harmonics of its own; any amp with that or a similar amount will be harsher and brighter (more 'solid state' sounding) than the input signal. This is because the ear interprets the higher ordered harmonics (5th and above) as harshness and brightness. Such amps may sound 'loud' as well, since the ear uses those harmonics to sense sound pressure, so they will impart a sense of loudness. A sound pressure level meter will show the truth of the matter.
So now you have a rough gauge of how amps will sound if you can get enough information about them; triode amps operating class A will be at one end of the 'smoothness' spectrum and class AB tetrode or pentode amps with push pull operation and a single-ended input, with feedback will be at the other end. However in some cases the latter will also be more detailed as they will have lower overall distortion. The one exception is that if you can find a balanced triode tube amp that lacks feedback you'll get both smoothness and detail, since this type of amp tends to have the greatest linearity overall (whether feedback is used or not- again, feedback will increase brightness and harshness).
One variable of all this is output impedance. Most speakers are designed to expect that the amp will make constant voltage regardless of the speaker load. For that you need a low output impedance and the only way you can get that is by adding feedback in a tube amp. So the speaker choice can have a dramatic effect on the results you get. A simple tip, regardless of the amp: Avoid Four Ohm Speakers. They will cause any amp to make more distortion and with tube amps also less power and if the amp employs an output transformer (most do but there are exceptions known as OTLs), a loss of low frequency bandwidth by as much as an octave. In a nutshell the wrong speaker can shoot down the good qualities of **any** tube amp! So you have to be careful about that if you want to get the most out of your amplifier dollar investment.