We don't know.
I hate to post a defeatest answer but I don't think it's at all clear what this smoke is or why it appears at this moment in the story. We see several other examples of Gringotts vaults being opened in the series and there's no green smoke mentioned on any of those occasions.
On the other occasions that Harry gets money out of Gringotts, somebody else does it for him or his trip to the vault isn't actually described in the narrative. However, in each of the four examples above, there is no mention given of the distinctive green smoke that appears in the Potter vault.
I can guess several reasons why this may be the case.
- Having Harry's wealth slowly materialise through a haze of smoke adds a touch of drama and heightens his sense of surprise at what he has (compared with what he was expecting to have). This lends emphasis to the 'poor orphan done good' theme.
- The green smoke might emenate from every vault on every opening, but JK Rowling neglected to take the effort to describe this every time someone went into Gringotts (or became tired of doing so).
The goblins of Gringotts take guarding their treasure seriously. It may well be the case that the Potters' vault had additional protections to it and that Griphook disabling these caused some green smoke to appear. (If this is true then it begs the question as to why the Flamel and Lestrange vaults didn't have this protection since they contained more valuable treasure - although the latter did have a blind dragon guarding it).
The Potters' vault hadn't been opened in a long time (ten years, at least). It may be that either wizarding money or the Gringotts vaults themselves give off this smoke over time. This would explain why Harry's vault isn't said to have any smoke coming out from it a year later.
- The Harry Potter Companion speculates that the smoke could have been a charm used by the goblins to protect the coins from corrosion (thanks to Valorum for the tip):
Since the green smoke was not toxic and Harry's coins appeared in perfect condition after 10 years underground, it may have been an anti-tarnish device as bronze and silver tend to go off if not polished by constant handling.
- We're not told the identity of the goblin who takes Harry and the Weasleys down to their vaults in Chamber of Secrets. It could be that the smoke was a byproduct of the way in which Griphook opens the vaults (although this is unlikely to be the true reason since he also opens the Flamel vault).
I'm afraid this doesn't add much more detail than that contained in the question itself. But, as JK Rowling hasn't addressed this to my knowledge, educated guesswork is really the best we can do.