Nothingness: I think white has a huge advantage - with perfect play, white cannot lose. Here's a proof.

Suppose a game is won for black (the second player) - that is, there is a strategy for black that always wins, regardless of how white plays. Then white starts by playing a random stone (this is never a disadvantage), and then adopting the strategy that wins the game for the second player. If that strategy requires placing a stone where white already had played a stone - white plays the stone at a random position. But that means white wins, contradicting the hypothesus that the game would be won for black. Ergo, there is no winning strategy for black if white doesn't make a mistake.

This 'proof' works for any game where placed stones don't move, and don't influence placement of other stones, and where having a stone at a certain position is never a hindrance. Examples outside PahTum include Five in Line, and Hex.