There are three things I can think of:
1) linearity- this is why triodes and ultralinear are more 3D, because they are more linear
2) bandwidth- phase shift can mess with your perception of the soundstage. In tube amps the only way to prevent this is by having enough bandwidth so that the audio passband is unaffected. So 10KHz and below is where the soundstage information exists; to prevent phase shift in this region you need 100KHz bandwidth. This is easier to get with triodes and ultralinear due to the reduced output impedance.
3) Noise is an issue. This can be caused by ground loops within the circuit, resulting in a noise floor that is made up of harmonic and inharmonic information masquerading as noise. So how the audio circuit is grounded can affect its ability to reproduce low level detail, which is important for the soundstage and depth in particular.
I should point something out though. Just because you have a 'triode' mode and 'ultralinear' mode does not say that the amp is doing either of those proper justice. This is because triodes usually have a lower output impedance, so the optimal load for them is different. They also have less gain, which will affect how much feedback is actually occurring. Bias points are different too. In a nutshell, unless the designer really did their homework, compromises will likely overshadow the results. So if I were you I would be hesitant about drawing conclusions from such a small sample size.
If a particular amp sounds good to you, that is what is important.
1) linearity- this is why triodes and ultralinear are more 3D, because they are more linear
2) bandwidth- phase shift can mess with your perception of the soundstage. In tube amps the only way to prevent this is by having enough bandwidth so that the audio passband is unaffected. So 10KHz and below is where the soundstage information exists; to prevent phase shift in this region you need 100KHz bandwidth. This is easier to get with triodes and ultralinear due to the reduced output impedance.
3) Noise is an issue. This can be caused by ground loops within the circuit, resulting in a noise floor that is made up of harmonic and inharmonic information masquerading as noise. So how the audio circuit is grounded can affect its ability to reproduce low level detail, which is important for the soundstage and depth in particular.
I should point something out though. Just because you have a 'triode' mode and 'ultralinear' mode does not say that the amp is doing either of those proper justice. This is because triodes usually have a lower output impedance, so the optimal load for them is different. They also have less gain, which will affect how much feedback is actually occurring. Bias points are different too. In a nutshell, unless the designer really did their homework, compromises will likely overshadow the results. So if I were you I would be hesitant about drawing conclusions from such a small sample size.
If a particular amp sounds good to you, that is what is important.