I interpret this statement as:
Some tube amps create lots of distortion and others may be hardly distort at all.
Given the amount of money people spend on this 'hobby', aren't both worth avoiding at all costs?
Well, when both the tube and SS distortion are well below the level of audibility, and the differences are a tiny tiny fraction of a percent, it doesn't really matter. The tubey sound has more to do with output impedance and FR than anything else, for instance the Bottlehead Crack has an output impedance of 120 ohm, and the Valhalla 2 has an output impedance of 3.5/14 ohms (low/high gain). With 3.5 ohms you're well within the realm of solid state amps, and even with 14 ohm, you're still very comfortably within the 1/10 rule for impedance matching. However, with the 120 ohm Crack, you're blasting well past the 1/10 rule (which would be 30 ohms for the HD800). Being below 1/10 or 1/8 means you start to lose some driver control, especially in the bass area. You also have the headphone FR changing a bit based on the headphone's impedance graph. It's not necessarily bad, just different, and some people prefer it. For instance, the Sennheiser hdvd 800, the amp Senn made to pair with the HD800 has an effective output impedance of 43 ohms, making the fraction slightly less than 1/7. Sennheiser is certainly capable of making a low output impedance amp, especially considering it's expensive and solid state. They chose that output impedance for a reason.
You also generally see a slight increase in bass and decrease in treble, this mixed with the reduced driver control is most (but likely not all) of what contributes to the "tubey" sound. It's more of a preference than anything else, and doesn't have a ton to do with distortion. Tube amps do more traditionally lean towards lower order and even harmonic distortion, which is known for being more pleasureable, while solid state amps are usually more even in their distortion, but that comes more in to play when you are intentionally distorting the amp (one of the reasons tube guitar amps are popular) because any decent amp won't have audible distortion until clipping.
There's also the whole tube rolling thing, which allows you to play with the signature a little bit.
But if you're looking for raw benefits of tubes, one big one is the ability to provide more power into higher impedance loads. Solid state amps are able to output the most power into lower impedance loads, while tubes often are able to output the most power into higher impedance loads. Just as an example, looking at Schiit amps specifically, the Valhalla 2 (800mW) provides more power into the HD800's than the hybrid Lyr 2 (660mW), the solid state Magni 2 Uber (320mW) and Asgard 2 (380mW). It's even pretty much even with the crazy powerful Mjolnir! (850mW). Each of these amps is rated as being more powerful than the Valhalla 2, but once you get to higher impedance headphones, the Valhalla 2 (the only pure tube amp Schiit makes) pulls ahead.
Of course, there are also people who believe tube amps sound better than solid state, and that measurements don't necessarily match perceived sound quality, but that area is a bit of a battle ground and I'd rather not get into it (plus, I'm actually not really on one side or the other).
And honestly, the component in my signal chain that produces the most noise and distortion is my damn ODAC, which measures great, but sometimes throws audible noise and even distortion into my signal chain (until I power cycle it). My tube amp doesn't.