Monday's NY Times ran a story (pg. A1) on the persistence of illegal numbers games in New York, despite the creation of the state-run lottery in 1980, which was supposed to wipe them out. The article noted that the payoff from the illegal games averages about 600-to-1, while the government version pays 500-to-1. A government gambling official explained that "legal games... face certain competitive disadvantages." He listed that they don't take 25 cent bets, they don't "let people bet on credit," and they report winnings to the tax authorities. The second item is not really true: the state lotto doesn't extend credit, but they certainly don't stop someone from borrowing money to play. In fact, the operators of illegal games who extend credit have a reason to try to ascertain that their customers aren't over-extending themselves, which the government games don't.
In any case, none of these reasons explain the lower payoff in the state version: they are all are circumstances that should make the state payout more, just as, for example, taxable bonds must offer higher yields than equivalent tax-free competitors. The lower payoff arises because the state isn't trying to compete with the illegal games by offering a better deal, but by threatening its competitors with violence.