I like both!
The ONLY thing I know that SS amps seem to do better 'generally' to me is bass. Otherwise, all bets are off. I've heard harsh and sterile tube amps and warm, buttery ss so it's tough to make sweeping statements.
Functionally, I like SS better... I don't have the 'nervosa' of wondering if a tube is going bad or if the bias is out of wack... or leaving it on overnight... or having someone bump into it... SS is just easier to own to me.
Tube amps have the advantage of infinitely adjusting the sound to meet your needs through tube swapping/rolling... To me that's not fun. I tend to pursue a specific sound and once I find it, I like it to stay that way.
I've found adding a tubed preamp to a SS amp is the way to go. (as have many, more experienced audiophiles) But that's me.
As to what the technical differences are, I'll let others explain. But those are my thoughts at the moment...
-C