Well, on a 70 tire you would need to cover more surface area to get the same lean then you would on a 65... Also the hight of your front tire would be slightly higher then the stock, so you would start leaning and the bike will start to dive over on it's own because you've increased the hight at the center... Your just playing around with what Kawasaki egineered, i would stick to the stock tire specs. It's like some people want to get a 190 tire for their 600, not all that great of an idea, you just messed with phisycs(sp), you would need to lean over harder....
To start, I've worked in the tire industry for about 5 years with 3 different tire companies, so I do know a little about tires.
120/70-17: obviously the 17 is your 17 inch rim, you can't change anything here. The 120 is your width, 120 mm wide. The 70 is your aspect ratio, basically the height of your tire. It tells you that the height is 70% of the width. 120 mm X .70 = 84 mm.
120/65-17: same deal. 120 mm X .65 = 78 mm.
A difference in tire height of 6 mm.
Is that a significant difference? I'm not sure. It doesn't really sound like that much. However, in my opinion, I think the theory presented by ZX Rider about a harder ride because of less are in the tire is bunk (no offense ZX)
It does affect the handling for sure, if your front height is suddenly reduced by 6mm, as explained by KawiT, it affects the Rake and Trail of the front suspension - the same as reducing your preload on the forks, also it depends a lot on the radius profile of the tread surface - usually the different aspect ratio will have a different profile - affecting the handling.
The dealers accidentally put a 50 profile instead of a 55 on the rear of our ZX10 when we replaced the rear tyre, and there was a definitive difference in handling, turning into a corner slower.
A lower profile would also mean that the tyre should be more stable on the rim therby you should be able to run slightly lower tyre pressures, but as you can see, and as mentioned above, if you are not racing the bike, and if you are not intending spending a lot of time and energy to experiment, stick to the stock size tyres.