We did a microphone comparison earlier this year and I just re-listened to it to refresh my memory. I wanted to compare two different microphones that differed only in the fact that one had a tube front-end and the other a FET front-end. I chose a RØDE NT-1000 and a RØDE NTK, These two mikes have the same HF-2 capsule, no transformer and the same transistor output circuit. They are physically very similar. The only real difference is that one is tube and the other is FET.
I compared the microphones on acoustic guitar and female vocal. The difference was subtle, but the NTK had a bit less high-end detail than the NT-1000. I preferred the sound of the NT-1000. The subtle sound of the fingers moving along the strings was much less in the NTK, in the acoustic guitar recording. That was really the only noticeable difference on guitar.
The vocalist, Danielle, has an amazingly clear and beautiful voice with no excessive sibilance. The NTK captured less of the high-frequency detail and airiness in her voice.
That said, the differences were subtle, and I can see that the NTK might be preferable on certain voices. Also, recording an acapella voice or a solo guitar is different than recording voices or instruments that have to fit into a mix.
So, in answer to the question, what I heard in the tube mike was softened transients and slightly rolled off high-end. I didn't notice any added harmonic richness.