Admission is the more common word in everyday English. It usually means permission to enter, the act of being accepted, or the fee required to get in. You might talk about admission to a museum, college admission, or free admission.
Admittance also relates to entry, but it is less common. It often sounds more formal and appears in signs or fixed phrases. For example, a sign may say, No admittance after 9 p.m. In many other contexts, native speakers would choose admission instead.
A simple way to remember the difference is this: if you mean permission, acceptance, or an entry fee, use admission. If you mean physical entry in a formal or sign like context, admittance can fit.
- Admission: The museum charges no admission.
- Admission: Her admission to the program came in April.
- Admittance: No admittance after 9 p.m.
When you are unsure, admission is usually the safer and more natural choice.