One of the major factors on whether a tube amplifier sounds different than solid state is how well the respective amplifiers can generate increased current into low impedance loads. If the amps are connected to speakers that have low sensitivity and a demanding impedance curve the two types of amps will likely sound quite different.
First example is driving speakers like Wilsons or Thiels that can dip to around 2 ohms over part of their frequency response. These speakers were designed and voiced to be powered by beefy amps that can double their power with each halving of the impedance. My Krell KSA 300s puts out 300 watts into 8 ohms and 2400 watts into 2 ohms. It is driving Thiel CS6 speakers that dip to around 2 ohms in the mid bass with a demanding phase angle. A typical tube amp will simply not be able to drive these speakers so that they have a flat frequency response. In the frequencies where the impedance drops to a low level, the amp cannot provide the current to drive the speakers.
Second example is driving speakers from a company like DeVore that are designed to have a flat impedance curve and higher sensitivity. There are also many horn speakers that have this characteristic. For these kinds of speakers a high quality tube amp and solid state amp will sound much more similar.
My point is that besides the pleasant sound that some people ascribe to tubes, there can be a large frequency response difference between amplifiers depending on what type of speakers you are driving. You will never see a pair of big Wilson speakers driven by an SET amp at an audio show. They are driven by monster amps typically from D'Agostino. They were designed and voiced using these amps and they were willing to accept the high current requirements to achieve the sound they want. One can argue that tubes have more "bloom" or whatever, but there can be large, measureable, frequency response differences depending on what amp you use to power speakers with a demanding impedance curve.