If your name is the same on both passports it will be enforced.
(When you will give your us passport in the passport control they will ask nice
for your israeli one).

Its not that they are going to arrest you..., But if you have the israeli one
make the biometric one, It will ease your in/out process...

I think there is some kind of data sharing between the countries
(For example my french passport is linked to the israeli one in the US system,
As they know i have to citizenship)