In the book Plunkitt of Tammany Hall William Riordan published many of George Washington Plunkitt's thoughts about government and about big city machines. In the link below, you can find the passage that explains the difference between honest and dishonest graft.
Honest graft is using your connections and knowledge as a government official to enrich yourself. It is essentially what we would now call "insider trading." Honest graft is when a goverment official goes out (for example) and buys up land because he knows a city project will need that land and he will be able to make a lot of money by buying the land now while no one else knows that it is about to be bought by the city. He can buy it cheap and then sell it at a higher price to the city.
Dishonest graft consists of doing things like blackmailing people who are doing illegal or semi-illegal things. It can also consist of actually taking money directly from the city treasury. It is more of what you would expect mobsters to do--things like forcing prostitutes to pay money to police in order to be allowed to work in a given area rather than being arrested.
Plunkitt compares "honest graft" to what is done on Wall Street. He sees it as a gentlemanly and quite acceptable way of using his poisition to enrich himself.
In Chapter 1 of Plunkitt of Tammany Hall, Plunkitt defines honest graft as "I seen my opportunities and I took 'em." In other words, if he hears, because of his political position, about a park that his political party is going to build in a neighborhood, he buys up the land around the park and makes a profit on it. Another type of honest graft is nepotism and favoritism. As Plunkitt says, "If I have a good thing to hand out in private life, I give it to a friend." In other words, he believes Tammany Hall should reward its friends.
On the other hand, he defines dishonest graft as overt rule breaking, such as stealing from the treasury, blackmailing people, or collaborating with gamblers and law breakers. As he says, "why should the Tammany leaders go into such dirty business, when there is so much honest graft lyin' around when they are in power?" In other words, Tammany Hall politicians don't need to engage in dishonest graft, as what he defines as honest graft is so lucrative.