My newborns pediatrician recommended I use Neosporin on his diaper area, but the kid kind I bought says it's not recommended for children under 2?

Lawyer effect. Over the counter medications often have disclaimers that are required by the companies legal department to help them avoid law suits if you use their product when a doctors visit should be considered. The full statement is not under 2 unless advised /under the care of a physician.You will see this on many things if you are a good label reader.It is fine to use.
Neosp is antibiotic. cream, & recommended to Rx or prevent infection in open sores. It is always preferred to treat topically rather than oral antibiotic, esp. in young infants. The amount of med absorbed through the skin is minimal, so safe for your baby. This is intended for short term use and regular diaper rash is treated with frequent diaper changes, wash with H2O & cover with zinc or petroleum based ointment.