The laws are the same for adults and minors other than the fact that youth offenders will see a mandatory of 6 months of driver license suspension for marijuana offenses that go beyond a ticket. This is usually for possession of heavy amounts of marijuana that can be seen as possible dealing of marijuana.
Possession of marijuana is treated the same for minors and adults apart from one small difference. A minor convicted of marijuana possession also gets a mandatory 6 month suspension of their drivers license. Also sale to a minor under 21 years by any party is a felony. Otherwise there is no other major differences.
In New York you can get medical marijuana on your own if you are age 18 or older. However if you are a minor and you are trying to get medical marijuana that you will need to bring a parent or legal guardian to the doctor with you. That way they can sign a waiver saying it is ok for you to get it.