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Here are most of the differences in play and in scoring between the New Zealand and Japanese rules.

Play it out

The biggest difference (to the Japanese rules) is that the New Zealand rules do not define what is dead. If there is any disagreement - play the position out and see what happens. Because of how we count, playing additional moves inside your territory does not affect the result.

False eyes in a seki.

The Japanese rule is 'No points are counted in a seki' The New Zealand rules do not mention sekis - stones that cannot be captured stay on the board and are counted, points inside a seki are counted too. It works out the same except when the seki includes false eyes. Vacant points surrounded by one colour only are counted for that colour. I think the New Zealand rules are more logical here.

The point at the end of the game

New Zealand rules counts stones on the board and points surrounded. Every stone is worth a point, if one player gets to play more stones than the other they will get extra points - assuming they do not waste them by playing inside their own territory.

This happens when Black gets to play the last move. If White plays the last move generally New Zealand rules will give the same result as Japanese rules. If Black plays last generally New Zealand rules will favour Black by one point compared to the Japanese rules. This seems bad, but surprisingly, it generally does not make a difference to who wins - provided the komi is odd:

Assuming a Komi of 5.5 and no oddities such as sekis...

Because there is an odd number of intersections, a margin of 1 in Black's favour will happen when Black and White have played an equal number of moves. When the margin is 2 (Japanese counting) Black will play last and be ahead by 3 with New Zealand rules counting... 

Margin (J) Margin (NZ) Winner (J) Winner (NZ)
1 1 W W
2 3 W W
3 3 W W
4 5 W W
5 5 W W
6 7 B B
7 7 B B
8 9 B B

When handicap stones are present, Black gets to count a point for each. (This is neither good nor bad, Handicaps are just worth slightly more under New Zealand rules than under Japanese rules.)

Ko at the end of the game - if you can win a Ko when there are no dame left (you take the Ko, your opponent passes, you fill the Ko) then you are one point better off under New Zealand rules than under Japanese rules. (This is not a bad thing, it's hard to engineer such a Ko)

Sometimes inside a seki there can be a dame that only one player can take (because if the other player takes it they die) The player who can take the dame does it at the end of the game, obtaining an extra point compared to Japanese scoring. (This is good - that point 'belongs' to the player who can take it - they should be rewarded)

Bent four in the corner

New Zealand rules do not define bent four in the corner. If a bent four exists and your opponent agrees it's dead - no problem. If not, demonstrate you can capture it. If you can't capture it -because your opponent has a Ko threat you cannot remove - then the bent four is not dead. (The New Zealand rules are definitely more logical here)

Suicide is allowed

The New Zealand rules make no mention of suicide, so it is allowed. (Mostly suicide is a very bad move - but so are many other moves, so that's not a reason to forbid it. Occasionally suicide can be used as a Ko threat - which is an excellent reason to allow it.)

Komi

We use 7 points, which allows for draws. (Why shouldn't it be possible to have a draw?)

Handicap stone positions

The New Zeland rules do not dictate the position of handicap stones. (Why should the position of the handicap stones be dictated? Allowing handicap players to experiance different opening corner positions is a good thing.)

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