Obsolete means "still around, but only marginally, because there is a newer, better variant". Extinct means "it's gone".
The text of the Obsolete quirk specifies mechs that are no longer in production for which compatible parts are not available. It doesn't actually require the mech to be superseded by a superior model, and has been used before for mechs that were simply extinct (such as the Rampage or Dragoon. Those weren't replaced by better hardware, they died out).
I can't even find a description for an "extinct" design quirk. And I went through TRO Golden Century, Battlemech Manual, Strategic Operation, the most recent Strategic Operation errata, and all of the primitive xtros just in case. Which book is it in?
Because just from the context of the name itself, it seems to either be doing a job already being done, or it's trying to say "don't use this mech, you would be WRONG". Which would be silly. You're not the boss of me, design quirk. I'ma throw a Vision Quest in a modern force just to spite you.