In most maps you start with more piles of gold than piles of stone.
Furthermore, each individual pile of gold has more gold (800) than each pile of stone has stone (350).
This means that you often have ~4x as much gold to mine as you do stone to mine.
Gold mining techs will end up saving ~4x as much "villlager time" as stone mining techs.
However, gold mining techs and stone mining tech cost the same.
The first gold mining tech saves you 1 hour of villager time, and the second tech, 45 minutes.
The first stone mining tech saves you 11 minutes of villager time, and the second tech, 2 minutes.
Note that this assumes you get the second mining tech before you begin mining any stone - if you have mined any of your starting stone then tech costs more time than it saves.
The second mining tech could be somewhat useful if:
you are mining more than just your starting stone (i.e. the "neutral" stones)
you need stone quickly (e.g. to build a castle to stop a push)
the enemy is pushing your stone and you need to get as much as possible before you lose control of it