The simple difference between the Fee NYC Apartments and No Fee NYC Apartments is not very complicated to understand. If you involved any agent in search for renting an apartment then you need to pay him some commission for searching a house for you then it said to be Fee Apartment. In case you did not take any help from any agent and you searched the apartment by yourself then you are not paying any commission to any one so it is called No Fee NYC Apartments.
If you are able to find an apartment by yourself and dealing directly with the landlord without any help from an agent then you will save a month rent or a 15% or year's rent as a commission. Sometimes landlords pay the agent (OP) because the agent works for the landlords or the building so the tenant need not to worry to pay him as he get paid by the owner by bringing a tenant to the apartment. The owner usually pays one month rent or 8.33% of year's rent. So even this is also considered as No Fee Apartments because you are not giving any money to agent he is paid by the owner as he works for them.
The owner of the building offers an OP (owner's payment) to an agent, because the agent advertises for the apartment as "No Fee NYC Apartments".
But the fact nobody gets it is, it may look nice and happy that you are able to find a no fee apartment but the landlords has no intention to save your money, after few months he will increase the rent as he don't want to loose his money to an agent.
To make it more clear about the concept of no fee apartment and fee apartment here it is suppose there are two similar buildings in NYC one is advertised as No Fee NYC Apartments and another one nothing. At no fee building the owner is paying to an agent because he works for him and the rent will be $2000 per month. But the second building considered as fee building here you have to pay some commission to an agent because he helped you find an apartment and the rent will be around $1,835 per month. In a year both the apartments costs same as $22,000 so it doesn't matter which building you are renting. What matters, is getting good apartment with full facilities.