"Contrast the Democratic party nominating rules with the Republican party nominating rules in New York. Which set of rules used advantages the front-runner more, the Democratic party rules or the Republican party rules?"

The Democratic Party primaries are all proportional in some form: If you get 30% of the votes, you get 30% (or so) of the delegates.Some Republican Party primaries, including New Jersey, are not proportional but winner-takes-all: If you get more votes than anyone else, you get all the delegates. Others have high thresholds for getting delegates, which has a similar effect.The general election is also winner-takes-all, for both parties.
The winner-takes-all system gives a substantial advantage to the front-runner relative to the proportional system. The threshold system gives a smaller but still significant advantage to the front-runner over a true proportional system.Let's take a look at the actual results for New Jersey, shall we?In the Democratic Party, Hillary Clinton received 63% of the vote and Bernie Sanders received 37% of the vote. Clinton got 85 delegates, which is, sure enough, 63% of the allocated delegates; and Sanders got 49 delegates, 37% of the delegates.In the Republican Party, Donald Trump got 81% of the votes, John Kasich got 13% of the votes, and Ted Cruz got 6% of the votes. But Donald Trump got 100% of the delegates, all 51 of them. Had they been allocated proportionally, Trump would have only gotten 41 delegates, while Kasich would get 7 and Cruz would get 3. Over many states, that difference can add up to a substantial advantage for the front-runner.You asked about New York specifically though.The New York primary is also proportional for both parties, but the Republican Party has a higher threshold for delegates. As a result, it behaves more like a winner-takes-all system.Hillary Clinton got 58% of the votes, and 56% of the delegates. Bernie Sanders got 42% of the vote, and 44% of the candidates. The rounding was actually slightly in favor of Sanders.While Donald Trump got 60.4% of the votes in New York, he got 89 delegates, which is 94% of the delegates! This happened because only John Kasich beat the threshold for getting any delegates, and only barely, getting 25% of the vote and only 6% of the delegates. Ted Cruz got 14% of the vote but no delegates at all.
It's not quite winner-takes-all like New Jersey, but it's quite close.
https://www.politico.com/mapdata-2016/2016-election/results/map/president/