Well, the plain voltage follower circuit is really no different in that regard. First and foremost, using a plain resistive attenuator will load the amp differently than a reactive speaker load, causing the amp to have a different type of, much flatter, frequency response. The voltage follower then just buffers the attenuated signal so that enough current can be pushed to low impedance loads. But the voltage follower - in its basic form will not colour the signal in any way worth of mentioning.I've tried building a work-a-like with a load resistor (30ohm), line out (facilitated by a pot wired as a voltage divider) into a standard poweramp, but it destroys the highs and dynamics, so that's a no-go.
If you didn't like a resistive attenuator coupled to a basic power amp, a good chance is that you won't like it coupled to plain voltage follower either because aside from a bit of voltage amplification in the first they really have no difference.
On that note, the UA's attenuator isn't a plain resistive one, based on the photos there seems to be an inductor in the circuit so the attenuator is likely more or less a simulating the complex reactive load of a loudspeaker.
The UA's voltage follower design in its basic form is pretty simple: It's just a push-pull emitter follower, biased by two diodes that are thermally coupled to the output transistors. It's pretty much the crudest example of such circuit. Such circuit topology also tolerates quite a lot of variation in component values. You can likely find some optimum of them but all in all, to get it working you don't really need to be a rocket scientist. Just putting it together with some proper layout will get it working.
Basic design is pretty much just two power transistors in emitter follower push-pull configuration. Emitter resistors R1 and R2 provide a bit of feedback to keep the transistors thermally stable. D1 and D2 provide the 0.6V voltage difference for emitter and base when the amp idles or approaches the crossover region, additionally they adjust this "bias" in interaction with temperature; hence you see them thermally coupled to the output transistors, basically by placing them to very close vicinity. R3 and R4 draw base current from the power supply, and again a wide tolerance for these components works. C1 and C2 are capacitive AC coupling from the attenuator network. Rail-to-rail voltage can be anything from about 40VDC to 120VDC, sometimes even more or less and basically depending on the pverall magnitude of attenuated signals and the power dissipation rating of the output transistors. It's a very basic design that you could improve and make more complex and better performer, but that is not something the UA would seem to do so let's just leave it at that.
I have naturally excluded the attenuator network from the schematic. It doesn't really matter in the overall picture: The emitter follower circuit is just a voltage follower for whatever input signal it gets.
I haven't depicted the rectifier/filtering circuit either but that's pretty basic too, as can fior example be seen in the following picture: Bridge rectifier configured for a dual rail supply, two filter caps. Center node is grounded.
Based on the photos, there seems to be a lot of variation in the UA versions, though. The one in the attached picture looks to be pretty much identical to my schematic. Others seem to include somekind of protection or at least a start up delaying circuit that drives a relay in the speaker line. Naturally the intention of such circuit is to prevent the startup "pop" you'd be hearing, or to disconnect the speaker line if there's DC in the output or if the current through the output transistors exceeds some threashold at which damage would begin to occur. Featuring some control for the attached fan is also a very possible scenario. Examples of such circuits can be found with Google.
In addition, if I interprete the photos correctly, some UA versions seem to feature a series resistor in the output coupled to the speaker; this resistor (typically having a value ranging from 1 to 4 ohms) would increase the voltage follower's output impedance, resulting into weaker damping and skewing the response like tube amplifiers with similar high output impedance tend to do. It would attenuate the overall signal a bit but additionally also enhance the response at speaker's resonant frequency and at higher frequencies. It might be the difference you are hearing. Not all UA design would portray that characteristic, though.
Some designs seem to also feature tone controls and one seems to hold something that could be a complete power amp board. Anyway, I'm not too keen on starting to guess and evaluate all what's inside them. Goes beyound the point. The basic version of the UA is indeed something very basic and pretty much follows the simple schema included in this post. But as stated earlier, it won't provide any different tone than just coupling the attenuator network to any linear power amp would.