Simply put though, I'm going to cover just a few basic differences between the two!
A 12ua7 Millet Max Hybrid Tube
Matrix M Stage- Solid State Amp
Most low impedance headphones require more current than they do voltage. In this case a solid state or Hybrid Tube amp is optimized for that current out put. So lower impedance headphones will benifit greatly on a solid state or hybrid tube! Low impedance headphones are most of what the average consumer market sells. Headphones like Sony and Audio Technica for example.
Tube amps, have higher voltage swings and not as much current, this allows headphone with high impedance [resistance to voltage] to be driven better due to the abundance of voltage. Again hyrbid tubes offer the current needed for low impedance cans and the voltage swing of tubes. In addition there Single Ended and Push Pull Tubes. Single ended are more limited in their power output but retain amazing details, where as Push Pull OTL tubes offer more variance in their power outputs but lack the detail of a Single End/
Now that being said, there are always exceptions. There are DOZENS of Tubes that can be used with an OTL Tube. Each of these having it's own combination of strengths and weakness.
A Solid State tube amp is most often harder to repair and replace parts for, where as a OTL and Hybrids pin some cases] are simple'r and easier to modify, repair and replace parts for. So for some one interested in the possibility of building their own amp, once again the are pros and cons to each.
In addition, going back to the huge selection of tubes. There is a more hobbyist approach to tube amps. As one amp can boast a variety of sound and detail levels, through multiple tubes! Almost how one Gas Top Stove can use a variety of different pots. In this food analogy, the Solid State amp would be like a Griddle. It does many things but only with one surface.
The key differences are tubes are more lush, as they produce more second harmonic and less odd 3'rd, 5'th ect harmonics. As we know sound travels in a wave, and as it's amplified these waves peak and trough differently. Ultimately even numbered harmonics, found in abundance in Tube Amps, sound more beautiful and natural. So the tube amp has a beauty to it, yet in contrast it can tend to loose instrument separation as well as sound stage imaging.
A Solid State, in contrast, does not offer as many of those natural harmonics but rather un rivaled detail and imaging. Solid States often contain better dynamics and positioning of sound. These are terms that describe how sound moves, how it changes.
So tubes offer a lusher softer, warmer sound. Where as a Solid State is a little more colder, better detail and a tighter punchier sound. As always, pros and cons, and a hybrid tube amp tries to combine the best of both worlds.
As a side note, bass has more impact with a solid state and more bloom a gentle'r kick with a tube. Depending on your own personal preference, bass can be enjoyable on both!
So, when using headphones that are more neutral Headphones that are know for detail, clarity and accuracy in imaging and dynamics, a tube will often add the warmth an beauty that may be missing, where as with headphones that boast lush mids, and smooth airy treble a solid state would improve the accuracy and detail of that can.
Personally, I use my Solid State for my beautiful lush Audio Technica W1000x, and I use my hybrid tube with my more open and spacious Byerdynamic DT 990 Pro 250 ohm.