That's a great question to which I can offer my opinion, because I wonder the same thing. It's a brand new home with killer art, but what does it offer the Dogs that the original home can't do better?
I think Pachinko nailed it though - "seems to work better in long term strategies than in [shorter term] ones" - I suspect that when facing a deck that cheats regularly (one with standard 3-value structure), this home sacrifices the short game for the long one. I think this manifests in three ways.
First, in how Justice doesn't require booting to produce bounties and thus can maneuver better in all phases of the game. Setting up early economy is key to winning a long game, as is positioning your dudes to disrupt your opponent's economy.
Second, there are a few key skilled Lawdogs and other cards that interact with bounty in a different way than just "bounty and ace" - there is both a mad scientist and a blessed that reduce opposing bounty to attach their respective hearts with greater ease. Or there are cards like Too Much Attention and Back Ways that exercise board control utilizing distributed bounties.
Third, doing the math independent of other card effects, the original home essentially says "+1 GR per turn" and the Justice home says "+1 GR per cheat and +2 GR per ace" - which means to me that the original home is a lot more steady whereas Justice could (theoretically) be leveraged to output greater GR rewards. Both require set up, and I think that looks different for each home.
So ultimately I think it boils down to a question of both quantity and quality that you need to ask yourself what your Lawdogs are trying to do. Quantity in how much bounty you reliably need on the board to enact your strategy - Quality in how exactly you will go about enacting your deck's strategy based on the available bounty.
That's my analysis so far. Cheers!