It is difficult to draw major distinctions between the two. You'll hear opinions that tubes are "warmer," but that isn't true for every tube amp. Not all solid state amps are cold and sterile, either. The sound you get depends on the way the circuit was designed and what parts were used. Further, the sound an amp makes depends on the headphones you're using it with. There are a few things like output impedance curves, power, etc. that will make an amp sound different depending on headphones.
More generally, I prefer tubes because they use simpler circuits, lend themselves to repair instead of replacement and because tubes have been in production for decades and likely will continue to be. When chips go out of production, remaining stock disappears and become impossible to find in a year or two. Anyone wanting to put them back into production faces a mountain of IP problems, too. Most tube designs are public domain today and can be reproduced without trouble. Tubes also used in guitar amps are plentiful and easy to find.
If you DIY, tubes can be easier to deal with. Solid state usually requires PCBs. Those are easy enough to stuff, but you can go point-to-point with tubes and adapt it into whatever case you have. Point-to-point is easy to repair, too. Tube amps also use fewer parts, so they're simpler to wire up.
Another difference is that it costs less to build a good solid state amp (and I do like some solid state amps, for the record) amp than a good tube amp. This is in the power supply, mostly. Solid state usually operates at 24V or less, making power transformers less costly. Most solid state amps are tightly regulated, too, with low cost parts.
Tubes need a high voltage power supply, usually between 250V and 400V. That takes bigger, better made transformers. If you want to regulate a tube power supply well, you'll need bigger, more costly capacitors and - hopefully - at least one choke. More chokes are better, though a choke costs almost as much as the power transformer. You can also improve things with tube rectifiers but that costs more, too. In very inexpensive tube amps, you'll find diode rectifiers, a cheap transformer and cheap, smaller capacitors. Those are usually inferior to the power supply found on a solid state amp that costs the same or less.
My advice is to go with solid state if you're on a budget. You'll get more amp for the money. If you want tubes - and they are wonderful - you have to pay more.
You can get great sound from either solid state or tubes. Just avoid the really cheap tube amps - you'll get the glowing tube, but not their full potential. The power supply
is the most important part and solid state does good power supplies for less money than you can with tubes.
