There's really no such thing as a "tube-driven tubescreamer".
The original has a pair of anti-phase diodes and a cap in a feedback loop around an IC. Plus a tubescreamer sounds like a tubescreamer (although it's not terribly different than the sound of a tweed Champ full-up, except for the heavy compression).
My personal opinion: The TS-808 is not really any better than any good-sounding tubescreamer.
I say that having owned a TS-808, and I was disappointed that it didn't live up to the hype. Essentially, it has the same compressed overdrive sound that all these types of distortion pedals have, eq differences aside.
The Fuultone Full-Drive 2 is a great pedal that can get both the compressed tubescreamer sound, as well as an uncompressed distortion sound (so you pick attack can cut through a band). There are also good modifications out there for run-of-the-mill tubescreamers. Tubenit did one (an changed IC, maybe some other changes) to a TS-10, which sounds every bit as good as the TS-808.
Most "tube based" pedals either starve the tube (use maybe 9-18v on the triode plate), or achieve their distortion sound with a standard clipped opamp or transistor, then run the resulting sound through a tube as a sort of eq. In my opinion, those don't live up to the hype, either. I had an Ibanez TubeKing way back when they first debuted; I returned it the following day because it sounded the same as the cheap sound-tank series Tubescreamer I had at the time.
In a whole other category are pedals like the Matchless Hotbox, which are really an preamp for a Marshall, but in a pedal. Those have their own challenges, but at least they claim "tube sound" honestly.