The difference really seems to come down to distortion. Tubes make more harmonic distortion, but it tends to be of the lower orders (2nd, 3rd and 4th). These are considered musical by the human ear/brain system.
Transistors in general make less distortion, but what they make tends to be of the higher ordered harmonics (5th and up). These are considered amusical by the human ear/brain system.
The brain uses the higher ordered harmonics as loudness cues, so if they are distorted you will get two results: the sound pressure will seem louder (and IMO/IME a stereo should not sound loud even if it is) and it will sound brighter, due to the human ear's extreme sensitivity to these harmonics.
The ear/brain system translates all distortions into tonality. So it converts the distortions of tubes into things that audiophiles describe as 'warmth', 'lushness', etc.
Otherwise both amps might measure flat on the bench, but the ear has a tipping point and will often favor tonality generated by distortion over actual frequency response errors.
There are ways to design tube amps to not make so much of the lower ordered harmonics without creating more of the higher harmonics (fully differential and balanced is one way). There are also ways to design transistor amps to make less of the higher ordered harmonics, and ways to make them make more of the lower ordered harmonics (single-ended, such as some of the Nelson Pass 'First Watt' amplifiers).
Once you understand that its all about how the amps distort (and I don't mean clip or overload, rather at normal listening levels) then you are a long ways closer to understanding what the differences are between them.
One other important difference is how the two technologies overload. Tube amplifiers overload gently and resist making higher ordered harmonics until really pressed hard. Transistor amps make lots of higher ordered harmonics as soon as they are clipped- and so its usually a bad idea to overload them as the clipping is amusical and irritating.
This is why you can often get by with less power with a tube amp than with solid state. It is also why guitar players tend to prefer tube amplifiers, as they often overload the amplifier as part of their 'sound'. Some of the best-sounding transistor-based guitar amps (Sunn comes to mind) were built in such a way that they made a fair amount of lower ordered harmonics- which is why they have a certain following as they are 'richer'.
Since transistor power is considerably cheaper than tube power, to get around the clipping problem a solid state amp will have a lot more power. The clipping problem must be dealt with as music tends to have powerful transients which makes it difficult to reproduce without distortion.